tag:blogger.com,1999:blog-30563183186100184942024-03-18T02:14:42.059-07:00Symbolab BlogMaking Math SimplerSymbolabhttp://www.blogger.com/profile/04610365055949018593noreply@blogger.comBlogger131125tag:blogger.com,1999:blog-3056318318610018494.post-66732860582227947012020-03-26T12:12:00.006-07:002022-06-08T07:31:19.914-07:00Advanced Math Solutions - Series Convergence Calculator, Series Ratio TestIn our Series blogs, we’ve gone over four types of series, Geometric, p, Alternating, and Telescoping, and their convergence tests. Now, we will focus on convergence tests for any type of infinite series, as long as they meet the tests’ criteria. We will start off with the Series Ratio Test.<br />
<br />
<b>What does the Series Ratio Test do?</b><br />
<br />
The goal of the Series Ratio Test is to determine if the series converges or diverges by evaluating the ratio of the general term of the series to its following term. The test determines if the ratio absolutely converges. A series absolutely convergences if the sum of the absolute value of the terms is finite. If there is absolute convergence, then there is convergence. This will make more sense, once you see the test and try out a few examples.<br />
<br />
Some caveats: The test will not determine what the series will converge to. The test may also result in inconclusive results.<br />
<br />
<b>What is the Series Ratio Test?</b><br />
<br />
Given a series, <span class="mathquill-embedded-latex font16">∑a_n</span> , determine L such that<br />
<br />
<span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{a_{n+1}}{a_n}\mid=L</span>,where <span class="mathquill-embedded-latex font16">a_n≠0</span><br />
If <span class="mathquill-embedded-latex font16">L<1</span>, then <span class="mathquill-embedded-latex font16">∑a_n</span> converges.<br />
If <span class="mathquill-embedded-latex font16">L>1</span>, then <span class="mathquill-embedded-latex font16">∑a_n</span> diverges.<br />
If <span class="mathquill-embedded-latex font16">L=1</span>, then the test is inconclusive.<br />
<br />
The test may seem pretty straight forward and simple, but determining what type of series to use this test on is not.<br />
<br />
<b>What types of series should the test be used on?</b><br />
<br />
While there is no straight forward answer to this question, this test is typically helpful when determining convergence for series with exponential functions or factorials.<br />
<br />
Now that we know what the series ratio test is, let’s see some examples of how it is used.<br />
<br />
First example (<span style="font-family: "times new roman" , serif; font-size: 12pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B2%5E%7Bn%7D%7D%7Bn!%7D?or=blog"><span face=""arial" , sans-serif" style="font-size: 11pt;">click here</span></a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=0}^∞\frac{2^n}{n!}</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Determine the limit of <span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{\frac{2^{n+1}}{(n+1)!}}{\frac{2^n}{n!}}\mid=lim_{n→∞} \mid\frac{2^{n+1}∙n!}{(n+1)!∙2^n }\mid</span><br />
<span class="mathquill-embedded-latex font16">=lim_{n→∞} \mid\frac{2∙n!}{(n+1)!}\mid</span><br />
<br />
(Note: <span class="mathquill-embedded-latex font16">2^{n+1} = 2^n ∙ 2^1</span> )<br />
<br />
<span class="mathquill-embedded-latex font16">=lim_{n→∞} \mid\frac{2}{(n+1) }\mid</span><br />
<br />
(Note: <span class="mathquill-embedded-latex font16">\frac{n!}{(n+m)!}=\frac{1}{(n+1)∙(n+2) ∙ ∙ ∙(n+m) }</span>)<br />
<br />
<span class="mathquill-embedded-latex font16">=0=L</span><br />
<br />
For detailed procedures on how to determine the limit, please see the step by step procedures in the link.<br />
<br />
<span style="white-space: pre;"> 2. </span>Given L, determine convergence<br />
<br />
Since <span class="mathquill-embedded-latex font16">L = 0</span> and is <span class="mathquill-embedded-latex font16">< 1</span>, the series converges.<br />
<br />
Next example (<span style="font-family: "times new roman" , serif; font-size: 12pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B6%5E%7Bn%7D%7D%7Bn%7D?or=blog"><span face=""arial" , sans-serif" style="font-size: 11pt;">click here)</span></a></span>:<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{6^n}{n}</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Determine the limit of <span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{\frac{6^{n+1}}{n+1}}{\frac{6^n}{n}}\mid=lim_{n→∞} \mid\frac{6^{n+1}∙n}{(n+1)∙6^n }\mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">=lim_{n→∞} \mid\frac{6n}{(n+1) }\mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">=6∙lim_{n→∞} \mid\frac{1}{(1+\frac{1}{n}) }\mid</span><br />
<br />
(Note: <span class="mathquill-embedded-latex font16">\frac{n}{n+1 }= \frac{1}{1+\frac{1}{n}}</span>)<br />
<br />
<span class="mathquill-embedded-latex font16">=6∙1=6=L</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Given L, determine convergence<br />
<br />
Since <span class="mathquill-embedded-latex font16">L = 6</span> and is <span class="mathquill-embedded-latex font16">> 1</span>, the series diverges.<br />
<br />
<br />
Next example (<span style="font-family: "times new roman" , serif; font-size: 12pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%20%20%7D%5Cfrac%7B2%5E%7B2%7D%5E%7Bn%7D%7D%7B3%5E%7B2n%7D%7D?or=blog"><span face=""arial" , sans-serif" style="font-size: 11pt;">click here</span></a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=0}^∞\frac{2^2n}{3^2n}</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Determine the limit of <span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{\frac{2^{2(n+1)}}{3^{2(n+1)}} }{\frac{2^2n}{3^2n} }\mid=lim_{n→∞} \mid\frac{(\frac{2}{3})^{2(n+1)}}{(\frac{2}{3})^2n} \mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">=lim_{n→∞} \mid(\frac{2}{3})^{2(n+1)-2n} \mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">=lim_{n→∞} \mid(\frac{2}{3})^2 \mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">=\frac{4}{9}=L</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Given L, determine convergence<br />
<br />
Since <span class="mathquill-embedded-latex font16">L = \frac{4}{9}</span> and is <span class="mathquill-embedded-latex font16">< 1</span>, the series converges.<br />
<br />
Last example:<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{n}</span><br />
<br />
<br />
<span style="white-space: pre;"> 1. </span>Determine the limit of <span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid</span><br />
<br />
<span class="mathquill-embedded-latex font16">lim_{n→∞} \mid\frac{\frac{1}{n+1}}{\frac{1}{n}}\mid=lim_{n→∞} \mid\frac{n}{n+1}\mid</span><br />
<br />
<br />
<span class="mathquill-embedded-latex font16">=lim_{n→∞} \mid\frac{1}{1+\frac{1}{n}}\mid</span><br />
<br />
(<span style="font-family: "times new roman" , serif; font-size: 12pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft(%5Cfrac%7Bn%7D%7Bn%2B1%20%7D%5Cright)?or=blog"><span face=""arial" , sans-serif" style="font-size: 11pt;">Click here</span></a></span> for the limit step by step procedures)<br />
<br />
<span class="mathquill-embedded-latex font16">=1=L</span><br />
<br />
<br />
<span style="white-space: pre;"> 2. </span>Given L, determine convergence<br />
<br />
Since <span class="mathquill-embedded-latex font16">L = 1</span>, the test is inconclusive.<br />
<br />
You may be wondering what to do if the ratio series test is inconclusive. In the next couple of blog posts, we will be discussing other convergence tests that can be used when the ratio test is inconclusive. For more practice on the Ratio Series Test, check out Symbolab’s <span style="font-family: "times new roman" , serif; font-size: 12pt;"><a href="https://www.symbolab.com/practice/series-practice?subTopic=Telescoping%20Series%20Test#area=main&subtopic=Ratio%20Test"><span face=""arial" , sans-serif" style="font-size: 11pt;">Practice</span></a></span>.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-8034383664860988772020-03-18T05:53:00.001-07:002022-06-08T07:50:59.849-07:00Advanced Math - Series Convergence Calculator, Telescoping Series TestLast <span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="http://blog.symbolab.com/2019/10/advanced-math-solutions-series.html"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">blog post,</span></a></span> we went over what an alternating series is and how to determine if it converges using the alternating series test. In this blog post, we will discuss another infinite series, the telescoping series, and how to determine if it converges using the telescoping series test.<br />
<br />
If it isn’t clear right away, telescoping is synonymous with the word collapsing. A <b>telescoping series</b> is a series where almost all the terms cancel with the preceding or following term leaving just the initial and final terms, i.e. a series that can be collapsed into a few terms.<br />
<br />
Let’s see what this looks like . . .<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{n(n+1)}= ∑_{n=1}^∞\frac{1}{n}-\frac{1}{n+1}</span><br />
<br />
<span class="mathquill-embedded-latex font16">= (1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+ ...+(\frac{1}{n}-\frac{1}{n+1})</span><br />
<br />
<span class="mathquill-embedded-latex font16">=1-\frac{1}{n+1}</span><br />
<br />
As you can see, we are able to cancel out all terms except the first and last.<br />
<br />
Now that we’ve discussed what a telescoping series is, let’s go over the telescoping series test.<br />
<br />
<b>Telescoping Series Test:</b><br />
<br />
For a finite upper boundary, <span class="mathquill-embedded-latex font16">∑_{n=k}^N(a_{n+1}-a_n )=a_{N+1 }-a_k </span><br />
For an infinite upper boundary, if <span class="mathquill-embedded-latex font16">a_n→0</span>*, then <span class="mathquill-embedded-latex font16">∑_{n=k}^∞(a_{n+1}-a_n )= -a_k </span><br />
*If <span class="mathquill-embedded-latex font16">a_n </span>doesn’t converge to 0, then the series diverges.<br />
<br />
In regards to infinite series, we will focus on the infinite upper boundary scenario. In order to use this test, you will need to manipulate the series formula to equal <span class="mathquill-embedded-latex font16">a_{n+1}-a_n</span> where you can easily identify what <span class="mathquill-embedded-latex font16">a_{n+1}</span> and <span class="mathquill-embedded-latex font16">a_n</span> are. Also, please note that <b>if you are able to manipulate the series in this form, you can confirm that you have a telescoping series</b>. With practice, this will come more naturally.<br />
<br />
Let’s see some examples to better understand.<br />
<br />
First example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B5%7D%7Bn%7D-%5Cfrac%7B5%7D%7Bn%2B1%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{5}{n}-\frac{5}{n+1}</span><br />
<br />
<br />
<span style="white-space: pre;"> 1. </span>Convert the series into the form <span class="mathquill-embedded-latex font16">a_{n+1}-a_n</span><br />
<br />
<span class="mathquill-embedded-latex font16">\frac{5}{n}-\frac{5}{n+1}= -\frac{5}{n+1}-(-\frac{5}{n})</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_{n+1}=-\frac{5}{n+1}</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{5}{n}</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Determine if <span class="mathquill-embedded-latex font16">a_n→0</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{5}{n}= -5(\frac{1}{n})</span><br />
<br />
Since <span class="mathquill-embedded-latex font16">\frac{1}{n}</span> converges to 0, <span class="mathquill-embedded-latex font16">-\frac{5}{n}</span> converges to 0.<br />
<br />
<span style="white-space: pre;"> 3. </span>Calculate <span class="mathquill-embedded-latex font16">-a_k</span><br />
<br />
<span class="mathquill-embedded-latex font16">k=1</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{5}{n}</span><br />
<br />
<span class="mathquill-embedded-latex font16">-a_k=-(-\frac{5}{1})=5</span><br />
The series converges to 5.<br />
<br />
Next example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B6%7D%7B%5Cleft(n%2B1%5Cright)%5Cleft(n%2B2%5Cright)%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>)<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{6}{(n+1)(n+2)}</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Convert the series into the form <span class="mathquill-embedded-latex font16">a_{n+1}-a_n</span><br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{6}{(n+1)(n+2)}= 6∙∑_{n=1}^∞\frac{1}{(n+1)(n+2)}</span><br />
<br />
<span class="mathquill-embedded-latex font16">\frac{1}{(n+1)(n+2)}= -(\frac{1}{n+2})-(-\frac{1}{n+1})</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_{n+1}=-\frac{1}{n+2}</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{1}{n+1}</span><br />
<br />
<br />
<span style="white-space: pre;"> 2. </span>Determine if <span class="mathquill-embedded-latex font16">a_n→0</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{1}{n+1}</span><br />
<br />
<span class="mathquill-embedded-latex font16">-\frac{1}{n+1} →0</span><br />
<br />
<span style="white-space: pre;"> 3. </span>Calculate <span class="mathquill-embedded-latex font16">-a_k</span><br />
<br />
<span class="mathquill-embedded-latex font16">k=1</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{1}{n+1}</span><br />
<br />
<span class="mathquill-embedded-latex font16">-a_k=-(-\frac{1}{1+1})=\frac{1}{2}</span><br />
<br />
<br />
<span class="mathquill-embedded-latex font16">6∙∑_{n=1}^∞\frac{1}{(n+1)(n+2)} =6∙\frac{1}{2}=3</span><br />
<br />
The series converges to 3.<br />
<br />
Last example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B1%7D%7B4n%5E%7B2%7D-1%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{4n^2-1}</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Convert the series into the form <span class="mathquill-embedded-latex font16">a_{n+1}-a_n</span><br />
<br />
<span class="mathquill-embedded-latex font16">\frac{1}{4n^2-1}=-(\frac{1}{2(2n+1)} )-(-\frac{1}{2(2n-1)})</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_{n+1}= -(\frac{1}{2(2n+1)} )</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{1}{2(2n-1)}</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Determine if <span class="mathquill-embedded-latex font16">a_n→0</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{1}{2(2n-1)} =-\frac{1}{4n-1}</span><br />
<br />
<span class="mathquill-embedded-latex font16">-\frac{1}{4n-1} →0</span><br />
<br />
<br />
<span style="white-space: pre;"> 3. </span>Calculate <span class="mathquill-embedded-latex font16">-a_k</span><br />
<br />
<span class="mathquill-embedded-latex font16">k=1</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=-\frac{1}{2(2n-1)}</span><br />
<br />
<span class="mathquill-embedded-latex font16">-a_k=-(-\frac{1}{2(2∙1-1)} )=\frac{1}{2}</span><br />
<br />
The series converges to <span class="mathquill-embedded-latex font16">\frac{1}{2}</span>.<br />
<br />
The trickiest part of this is manipulating the series formula into <span class="mathquill-embedded-latex font16">a_{n+1}-a_n</span>. Once you’re able to do this, the rest should be pretty simple. The key thing to remember about a telescoping series is that all the terms will cancel out, except the first and last term.<br />
<br />
For more help on telescoping series, check out Symbolab’s <span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US;"><a href="https://www.symbolab.com/practice/series-practice?subTopic=Alternating%20Series%20Test#area=main&subtopic=Telescoping%20Series%20Test"><span style="font-family: "Arial",sans-serif; font-size: 11.0pt; mso-bidi-font-weight: bold;">Practice</span></a></span>. Next blog post, I’ll go over the convergence test for a radio series.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-35850182888912157522019-10-21T09:52:00.002-07:002022-06-08T07:53:01.025-07:00Advanced Math Solutions - Series Convergence Calculator, Alternating Series TestLast blog post, we discussed how to determine if an infinite p-series converges using the p-series test. In this blog post, we will discuss how to determine if an infinite alternating series converges using the alternating series test.<br />
<br />
An <b>alternating series</b> is a series in the form <span class="mathquill-embedded-latex font16">∑_{n=0}^∞(-1)^n∙a_n</span> or <span class="mathquill-embedded-latex font16">∑_{n=0}^∞(-1)^{n-1}∙a_n</span>, where <span class="mathquill-embedded-latex font16">a_n>0</span> for all <span class="mathquill-embedded-latex font16">n</span>. As you can see, the alternating series got its name from its terms that alternate between positive and negative values.<br />
<br />
In some problems, you may come across alternating sign expressions that aren’t listed above. These various sign expressions can be simplified to equal the ones listed above. For example:<br />
<br />
<span class="mathquill-embedded-latex font16"> (-1)^{n+1}=(-1)^{n-1}∙(-1)^2=(-1)^{n-1}∙1=(-1)^{n-1}</span><br />
<br />
Now that we know what an alternating series is, let’s discuss how to determine if the series converges, using the alternating series test.<br />
<br />
<b>Alternating Series Test:</b><br />
<br />
An alternating series converges if all of the following conditions are met:<br />
<span style="white-space: pre;"> 1. </span><span class="mathquill-embedded-latex font16">a_n>0</span> for all <span class="mathquill-embedded-latex font16">n</span><br />
<blockquote class="tr_bq">
<ul>
<li><span class="mathquill-embedded-latex font16">a_n</span> is positive</li>
</ul>
</blockquote>
<span style="white-space: pre;"> 2. </span><span class="mathquill-embedded-latex font16">a_n>a_(n+1)</span> for all <span class="mathquill-embedded-latex font16">n≥N</span>,where <span class="mathquill-embedded-latex font16">N</span> is some integer<br />
<blockquote class="tr_bq">
<ul>
<li><span class="mathquill-embedded-latex font16">a_n</span> is always decreasing</li>
</ul>
</blockquote>
<span style="white-space: pre;"> 3. </span><span class="mathquill-embedded-latex font16">lim_{n→∞} a_n=0</span><br />
<br />
If an alternating series fails to meet one of the conditions, it doesn’t mean the series diverges. There are other tests that can be used to determine divergence.<br />
<br />
Let’s see some examples of how to use the alternating series test to help you better understand.<br />
<br />
First example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/convergence%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft(-1%5Cright)%5E%7Bn%2B1%7D%7D%7B%5Csqrt%7Bn%2B1%7D%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{(-1)^{n+1}}{\sqrt{n+1}}</span><br />
<span style="white-space: pre;"> 1. </span>Is <span class="mathquill-embedded-latex font16">a_n>0</span> for all <span class="mathquill-embedded-latex font16">n</span><br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{(-1)^{n+1}}{\sqrt{n+1}}=∑_{n=1}^∞(-1)^{n+1}∙\frac{1}{\sqrt{n+1}}</span><br />
<br />
<span class="mathquill-embedded-latex font16">a_n=\frac{1}{\sqrt{n+1}}>0</span> for all n<br />
<br />
<span style="white-space: pre;"> 2. </span>Is <span class="mathquill-embedded-latex font16">a_n>a_(n+1)</span> for all <span class="mathquill-embedded-latex font16">n≥N</span>,where <span class="mathquill-embedded-latex font16">N</span> is some integer?<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{\sqrt{n+1}}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+ ...</span><br />
<span class="mathquill-embedded-latex font16">\frac{1}{\sqrt{2}}>\frac{1}{\sqrt{3}}</span> and <span class="mathquill-embedded-latex font16">\frac{1}{\sqrt{3}}>\frac{1}{\sqrt{4}}</span> and so on<br />
<br />
<span class="mathquill-embedded-latex font16">a_n>a_(n+1)</span> for all <span class="mathquill-embedded-latex font16">n≥1</span><br />
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<span style="white-space: pre;"> 3. </span>Is <span class="mathquill-embedded-latex font16">lim_{n→∞} a_n=0</span>?<br />
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Let’s use this property to determine the limit of <span class="mathquill-embedded-latex font16">a_n</span>:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFjPvgXVeZHQcc3B9KSdgNgT0Pr-JViK4FEAosZ0eGgthQHQT3PPnZGN73g1nSZbJjeEMzlJd4rdcEuHKDOV1P3RD86wZgvm5GQaaBrCBLxpVhnBSL71joWKiZJNd9TPYiZWT6BP88y4M/s1600/picsym1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="100" data-original-width="692" height="55" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFjPvgXVeZHQcc3B9KSdgNgT0Pr-JViK4FEAosZ0eGgthQHQT3PPnZGN73g1nSZbJjeEMzlJd4rdcEuHKDOV1P3RD86wZgvm5GQaaBrCBLxpVhnBSL71joWKiZJNd9TPYiZWT6BP88y4M/s400/picsym1.png" width="400" /></a></div>
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<span class="mathquill-embedded-latex font16">lim_{n→∞} \frac{1}{\sqrt{n+1}}=\frac{lim_{n→∞} 1}{lim_{n→∞} \sqrt{n+1}}=\frac{1}{∞}=0</span><br />
<br />
<span style="white-space: pre;"> 4. </span>Are all three conditions met?<br />
<br />
Yes, the series converges.<br />
<br />
Next example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/convergence%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft(-1%5Cright)%5E%7Bn%7D%5Cleft(n%5Cright)%7D%7Bn%5E%7B2%7D%2B2%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{(-1)^n (n)}{n^2+2}</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Is <span class="mathquill-embedded-latex font16">a_n>0</span> for all <span class="mathquill-embedded-latex font16">n</span>?<br />
<br />
<span class="mathquill-embedded-latex font16">a_n=\frac{n}{n^2+2}>0</span> for all <span class="mathquill-embedded-latex font16">n</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Is <span class="mathquill-embedded-latex font16">a_n>a_{n+1}</span> for all <span class="mathquill-embedded-latex font16">n≥N</span>,where <span class="mathquill-embedded-latex font16">N</span> is some integer?<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{n}{n^2+2}=\frac{1}{3}+\frac{2}{6}+\frac{3}{11}+\frac{4}{18} ...=\frac{1}{3}+\frac{1}{3}+\frac{3}{11}+\frac{2}{9} ...</span><br />
<br />
<span class="mathquill-embedded-latex font16">\frac{1}{3}=\frac{1}{3}</span>, <span class="mathquill-embedded-latex font16">\frac{1}{3}>\frac{3}{11}</span>, <span class="mathquill-embedded-latex font16">\frac{3}{11}>\frac{2}{9}</span>, and so on<br />
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<span class="mathquill-embedded-latex font16">a_n>a_(n+1)</span> for all <span class="mathquill-embedded-latex font16">n≥2</span><br />
<br />
<span style="white-space: pre;"> 3. </span>Is <span class="mathquill-embedded-latex font16">lim_{n→∞} a_n=0</span>?<br />
<br />
<span class="mathquill-embedded-latex font16"> lim_{n→∞} \frac{n}{n^2+1}=lim_{n→∞} \frac{\frac{1}{n}}{1+\frac{1}{n^2}}=\frac{lim_{n→∞} \frac{1}{n}}{lim_{n→∞} 1+\frac{1}{n^2}}= \frac{0}{1+0}=0</span><br />
<br />
<span style="white-space: pre;"> 4. </span>Are all three conditions met?<br />
<br />
Yes, the series converges.<br />
<br />
Last example (<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US;"><span style="font-family: "Arial",sans-serif; font-size: 11.0pt; mso-bidi-font-weight: bold;"><a href="https://www.symbolab.com/solver/step-by-step/convergence%20%5Csum_%7Bn%3D2%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft(-1%5Cright)%5E%7Bn%2B2%7D%7D%7Bln%5Cleft(n%5Cright)%7D?or=blog">click here</a>)</span></span>:<br />
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<span class="mathquill-embedded-latex font16">∑_{n=2}^∞\frac{(-1)^{n+2}}{ln(n)}</span><br />
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As discussed in the beginning, we can see a different alternating sign expression that wasn’t listed. However, we can simplify this expression to equal one that was listed.<br />
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<span class="mathquill-embedded-latex font16">(-1)^{n+2}=(-1)^n*(-1)^2=(-1)^n</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Is <span class="mathquill-embedded-latex font16">a_n>0</span> for all <span class="mathquill-embedded-latex font16">n</span>?<br />
<br />
<span class="mathquill-embedded-latex font16">a_n=\frac{1}{ln(n)}>0</span> for all <span class="mathquill-embedded-latex font16">n</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Is <span class="mathquill-embedded-latex font16">a_n>a_{n+1}</span> for all <span class="mathquill-embedded-latex font16">n≥N</span>,where <span class="mathquill-embedded-latex font16">N</span> is some integer?<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=2}^∞\frac{1}{ln(n)}=\frac{1}{ln(2)}+\frac{1}{ln(3)}+\frac{1}{ln(4)} ...</span><br />
<br />
<span class="mathquill-embedded-latex font16">\frac{1}{ln(2)}>\frac{1}{ln(3)}</span>, <span class="mathquill-embedded-latex font16">\frac{1}{ln(3)}>\frac{1}{ln(4)}</span>, and so on<br />
<br />
<span class="mathquill-embedded-latex font16">a_n>a_{n+1}</span> for all <span class="mathquill-embedded-latex font16">n≥2</span><br />
<br />
<span style="white-space: pre;"> 3. </span>Is <span class="mathquill-embedded-latex font16">lim_{n→∞} a_n=0</span>?<br />
<br />
<span class="mathquill-embedded-latex font16">lim_{n→∞} \frac{1}{ln(n)}=\frac{lim_{n→∞} 1}{lim_{n→∞} ln(n)}=\frac{1}{∞}=0</span><br />
<br />
<span style="white-space: pre;"> 4. </span>Are all three conditions met?<br />
<br />
Yes, the series converges.<br />
<br />
The alternating series test has a lot of parts to it, but as long as you remember the three conditions, you’ll be able to master this topic! For more help or practice on the alternating series test, check out Symbolab’s <span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/practice/series-practice?subTopic=Geometric%20Series%20Test"><span style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt;">Practice</span></a></span>. Next blog post, I’ll go over the convergence test for telescoping series.<br />
<br />
Until next time,<br />
<br />
LeahUnknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-26658074861857842232019-07-24T07:54:00.002-07:002022-06-08T07:56:43.743-07:00High School Math Solutions - Series Convergence Calculator, p-Series TestLast <span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="http://blog.symbolab.com/2019/05/advanced-math-solutions-series.html?fbclid=IwAR2e_njWqY56xygOv-GhH9JG0spQt9puO-B0rOv5yE97FMvCqClDq8GRxgM"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">blog post</span></a></span>, we discussed what an infinite series is and how to determine if an infinite series converges using the geometric series test. In this blog post, we will discuss how to determine if an infinite series converges using the p-series test.<br />
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A p-series is a series of the form<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{n^p}</span> , where p is a constant power.<br />
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Here is an example of a p-series:<br />
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<span class="mathquill-embedded-latex font16">1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+ ...=\frac{1}{1^2} +\frac{1}{2^2} +\frac{1}{3^2} +\frac{1}{4^2} + ...=∑_{n=1}^∞\frac{1}{n^2}</span><br />
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So, how do we determine if the sum of a p-series converges to a finite number or diverges to an infinite number? We use the p-series test!<br />
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The following is the p-series test:<br />
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If the series is of the form <span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{n^p}</span> , where <span class="mathquill-embedded-latex font16">p>0</span>, then<br />
<span style="white-space: pre;"> </span>If <span class="mathquill-embedded-latex font16">p>1</span>, then the series converges.<br />
<span style="white-space: pre;"> </span>If <span class="mathquill-embedded-latex font16">0≤p<1</span>, then the series diverges.<br />
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Unlike the geometric test, we are only able to determine whether the series diverges or converges and not what the series converges to, if it converges.<br />
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The p-series test is fairly simple, useful, and easy to remember.<br />
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Let’s see some examples of how to use it.<br />
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First example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B1%7D%7B%5Csqrt%7Bn%7D%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>):<br />
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<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{\sqrt{n}}</span><br />
<span style="white-space: pre;"> 1. </span>Determine the value of p<br />
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<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{1}{\sqrt{n}}= ∑_{n=1}^∞\frac{1}{n^{\frac{1}{2}}}</span><br />
<span class="mathquill-embedded-latex font16"><br /></span>
<span class="mathquill-embedded-latex font16">p= \frac{1}{2}</span><br />
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<span style="white-space: pre;"> 2. </span>Determine whether the series converges or diverges<br />
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Since <span class="mathquill-embedded-latex font16">p= \frac{1}{2}</span> and therefore <span class="mathquill-embedded-latex font16">0≤p<1</span>, the series diverges.<br />
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Next example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7Bn%5E%7B2%7D%7D%7B%5E%7Bn%5E6%7D%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>):<br />
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<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{n^2}{n^6}</span><br />
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<span style="white-space: pre;"> 1. </span>Determine the value of p<br />
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<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{n^2}{n^6} = ∑_{n=1}^∞\frac{1}{n^{6-2}} = ∑_{n=1}^∞\frac{1}{n^4}</span><br />
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<span class="mathquill-embedded-latex font16">p=4</span><br />
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In this step, I used the following exponent rule: <span class="mathquill-embedded-latex font16">\frac{x^a}{x^b} =\frac{1}{x^{b-a}}</span><br />
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<span style="white-space: pre;"> 2. </span>Determine whether the series converges or diverges<br />
<br />
Since <span class="mathquill-embedded-latex font16">p=4</span> and therefore <span class="mathquill-embedded-latex font16">p>1</span>, the series converges.<br />
<br />
Last example (<span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7Bcos%5E%7B2%7D%5Cleft(n%5Cright)%2Bsin%5E%7B2%7D%5Cleft(n%5Cright)%7D%7B%5E%7Bn%5E2%7D%7D?or=blog"><span style="font-family: "arial" , sans-serif; font-size: 11.0pt;">click here</span></a></span>):<br />
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<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{cos^2(n)+sin^2(n)}{n^2}</span><br />
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<span style="white-space: pre;"> 1. </span>Determine the value of p<br />
<br />
<span class="mathquill-embedded-latex font16">∑_{n=1}^∞\frac{cos^2(n)+sin^2(n)}{n^2} = ∑_{n=1}^∞\frac{1}{n^2}</span><br />
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<span class="mathquill-embedded-latex font16">p=2</span><br />
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<span style="white-space: pre;"> </span>In this step, I used the following trigonometric identity: <span class="mathquill-embedded-latex font16">sin^2(x)+cos^2(x)=1</span><br />
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<span style="white-space: pre;"> 2. </span>Determine whether the series converges or diverges<br />
<br />
Since <span class="mathquill-embedded-latex font16">p=4</span> and therefore <span class="mathquill-embedded-latex font16">p>1</span>, the series converges.<br />
<br />
The p-series test is pretty straightforward, helpful, and not too difficult. For more help or practice on the p-series test, check out Symbolab’s <span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><a href="https://www.symbolab.com/practice/series-practice?subTopic=Geometric%20Series%20Test"><span style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt;">Practice</span></a></span>. Next blog post, I’ll go over the convergence test for alternating series.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-80604204611915062442019-05-13T06:00:00.001-07:002022-06-08T07:56:44.902-07:00Advanced Math Solutions - Series Convergence Calculator, Geometric SeriesSeries are an important part of Calculus. In this next series of blog posts, I will be discussing infinite series and how to determine if they converge or diverge.<br />
<br />
For a refresher:<br />
<br />
A series is the sum of a list of terms that are generated with a pattern. A series is denoted with a summation symbol. An infinite series is a series that has an infinite number of terms being added together.<br />
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Here is an example of an infinite series:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnOK_-r736lbfuD6yQ13oDBxCVB0J5yqTWSF6joeQ4Pm5x7Ay6fgj3f1qWCcqbqV7ripqY8Qeebs2NSy7DLVoOfkWb6H7tj0WSFS1hIuhekCnIznKK47TRxPsjqvdsU6UslgUtt8ZNrik/s1600/picsym1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="40" data-original-width="457" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnOK_-r736lbfuD6yQ13oDBxCVB0J5yqTWSF6joeQ4Pm5x7Ay6fgj3f1qWCcqbqV7ripqY8Qeebs2NSy7DLVoOfkWb6H7tj0WSFS1hIuhekCnIznKK47TRxPsjqvdsU6UslgUtt8ZNrik/s1600/picsym1.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD7TT7bJ3_tuluR0WDYAfh07fzp0HpGQdJN1s95ALvw6NLgokBzn7YKD9Zh15O2zXpJYXgjxHLYopbnjV70L9Ms45-Y4O_3KnwNhpOvBNfI-owhbWKjVsBq3ZEmQ2IGJCqkyRA0EBJyzo/s1600/picsym2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="42" data-original-width="175" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD7TT7bJ3_tuluR0WDYAfh07fzp0HpGQdJN1s95ALvw6NLgokBzn7YKD9Zh15O2zXpJYXgjxHLYopbnjV70L9Ms45-Y4O_3KnwNhpOvBNfI-owhbWKjVsBq3ZEmQ2IGJCqkyRA0EBJyzo/s1600/picsym2.png" /></a></div>
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With infinite series, it can be hard to determine if the series converges or diverges. Luckily, there are convergence tests to help us determine this!<br />
<br />
In this blog post, I will go over the convergence test for geometric series, a type of infinite series.<br />
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A geometric series is a series that has a constant ratio between successive terms. A visualization of this will help you better understand.<br />
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Here’s a geometric series:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2C7ANqCs03R7r0PkL_iqvh_MGkmDtWGtuSOdFE6dcOfRTRqAWtOi-d4D8ZvaLjDCUlV9uT7ZDZ2ISHK0RHkkpFZf8v1eqty2WFZ3vrWwjW-4k0aGcx70_EwoZcSV3HNQyW3G8VX32HZk/s1600/picsym3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="46" data-original-width="193" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2C7ANqCs03R7r0PkL_iqvh_MGkmDtWGtuSOdFE6dcOfRTRqAWtOi-d4D8ZvaLjDCUlV9uT7ZDZ2ISHK0RHkkpFZf8v1eqty2WFZ3vrWwjW-4k0aGcx70_EwoZcSV3HNQyW3G8VX32HZk/s1600/picsym3.png" /></a></div>
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In this series, each following term is the product of the prior term and ⅓.<br />
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We can rewrite this geometric series using the summation notation.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTYnJfzA9XijFRup_sEZAPl4OaMte_vC6-0slyUyVE2ShnMpUaiVQkZHLPKqIBHzrTwjq1EaJMJTFEUgBXZj0DCdjkQrrKZsNBXrmjbA4FoZERpuez6jhRPX3xs5YV7gXu8mOYmOyiS1E/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="47" data-original-width="315" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTYnJfzA9XijFRup_sEZAPl4OaMte_vC6-0slyUyVE2ShnMpUaiVQkZHLPKqIBHzrTwjq1EaJMJTFEUgBXZj0DCdjkQrrKZsNBXrmjbA4FoZERpuez6jhRPX3xs5YV7gXu8mOYmOyiS1E/s1600/picsym4.png" /></a></div>
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In order to determine if a geometric series diverges or converges, you’ll need to follow and remember the following test/rule:<br />
<br />
If the series is of the form <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhU82HJunIXdrOk2eVEWZ84EbYfhZbGA6qqXGivDj30xu9rGyZV8mLiFwNKIxDbK1ISoBb1lW2KrlaNJVUz-hvdaK0qNEKlrflqhT1G2XPsp5jNHNvF3LScC7NEEhDFEqrvAq_PSfbr__k/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="37" data-original-width="92" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhU82HJunIXdrOk2eVEWZ84EbYfhZbGA6qqXGivDj30xu9rGyZV8mLiFwNKIxDbK1ISoBb1lW2KrlaNJVUz-hvdaK0qNEKlrflqhT1G2XPsp5jNHNvF3LScC7NEEhDFEqrvAq_PSfbr__k/s1600/picsym5.png" /></a>,<br />
<br />
if |r|<1, then the geometric series converges to<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1nISsvEYI-U_ZB9iqGv_RqoIjY4RywbLey2wIRk4FuWBPOB4TYpMhOW1w77NfLT7_0x4RWigR9mbhu-DHYUxV0rr1E6nUUAGIDEjyqc6YSvrQ4TBLLVjGKwk73dEskD_j2PTTD9KHRv4/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="42" data-original-width="46" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1nISsvEYI-U_ZB9iqGv_RqoIjY4RywbLey2wIRk4FuWBPOB4TYpMhOW1w77NfLT7_0x4RWigR9mbhu-DHYUxV0rr1E6nUUAGIDEjyqc6YSvrQ4TBLLVjGKwk73dEskD_j2PTTD9KHRv4/s1600/picsym6.png" /></a><br />
if |r|≥1, then the geometric series diverges<br />
<br />
Let’s see some examples to better understand.<br />
<br />
First example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B1%7D%7B5%5E%7Bn%7D%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihF45494dqeb_Ev7zOVwSO58iCp786VRXYv6WOWDygAOLPfjuLzEm6W0orJaKhv562dAPx_trm8N-maRn_oeCtWRphfft5wydVHoyCMJOXuGnnDXa8JFprQT3o4AMPTWr6GnKTROXbgqI/s1600/picsym7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="60" data-original-width="98" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihF45494dqeb_Ev7zOVwSO58iCp786VRXYv6WOWDygAOLPfjuLzEm6W0orJaKhv562dAPx_trm8N-maRn_oeCtWRphfft5wydVHoyCMJOXuGnnDXa8JFprQT3o4AMPTWr6GnKTROXbgqI/s1600/picsym7.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Reference the geometric series convergence test<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkfdEkltxSRQr3erdjsUHriO0KUCSPsg2UoONmj4pEFm-V17PkmTJzStHX3RGEKQt_c5_9tF0MGKpU0jhR9XWInBUP2ucqvj34rX-0cAz4Jx0tk5wScZCjsxkdo8ZmWcU6v7bV6SDvKOg/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="469" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkfdEkltxSRQr3erdjsUHriO0KUCSPsg2UoONmj4pEFm-V17PkmTJzStHX3RGEKQt_c5_9tF0MGKpU0jhR9XWInBUP2ucqvj34rX-0cAz4Jx0tk5wScZCjsxkdo8ZmWcU6v7bV6SDvKOg/s1600/picsym8.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Determine the value of r<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPJWHvu1itBGskNOQ9tIEKxdy_1YMm6SscOBgIo_6c-8Fc_yCcoM79EZd7pmBoDHg9scT2ge2yCNzfMeXi1lHILEqoQY_fbtfRaW4jKPChZd_f9UQKsomV-kAhHaTv1pZNIMP7_iBtXxM/s1600/picsym9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="57" data-original-width="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPJWHvu1itBGskNOQ9tIEKxdy_1YMm6SscOBgIo_6c-8Fc_yCcoM79EZd7pmBoDHg9scT2ge2yCNzfMeXi1lHILEqoQY_fbtfRaW4jKPChZd_f9UQKsomV-kAhHaTv1pZNIMP7_iBtXxM/s1600/picsym9.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLj54fsDW96jaV00p2PogH1yOhHhlsxWyDpCP_B4zHpkWGeFgaG0PvClY083pnLiyb3_SlV-TsKdz5mIQiV1UYYRkJF0cC9vyFjB8r383KGGgrpilcYZ3ncIpGrFZaZx0_0fKIBcWU5B0/s1600/picsym10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="55" data-original-width="64" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLj54fsDW96jaV00p2PogH1yOhHhlsxWyDpCP_B4zHpkWGeFgaG0PvClY083pnLiyb3_SlV-TsKdz5mIQiV1UYYRkJF0cC9vyFjB8r383KGGgrpilcYZ3ncIpGrFZaZx0_0fKIBcWU5B0/s1600/picsym10.png" /></a></div>
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<span style="white-space: pre;"> 3. </span>Determine if the series converges or diverges<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOFeSbuo9gOq7Rfi66ER-MUydbYJJIZGLzCWghnQ11lirUUHzAKIbNP-Ilaaxe4LE5p2Kp2_mq2SGdaCIAtKxFS87qxtnmhMxmrYto3iirjBeI5Iy214XILH8PuqGmmdYXzcMsLdzGqJo/s1600/picsym11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="46" data-original-width="121" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOFeSbuo9gOq7Rfi66ER-MUydbYJJIZGLzCWghnQ11lirUUHzAKIbNP-Ilaaxe4LE5p2Kp2_mq2SGdaCIAtKxFS87qxtnmhMxmrYto3iirjBeI5Iy214XILH8PuqGmmdYXzcMsLdzGqJo/s1600/picsym11.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNUNzS4xVuPnl070MuXJTnuUikU6c5ZliA3bioTx7aEcQbTw3afxWT_1MfogaU4CNrsK6pEkAB_JslnSzPp1MJxBt2qMvTLHTqe7NxDcQ_il1yKfRlj4dUNKQaGzXKweuauhO0a0MNvE4/s1600/picsym12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="103" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNUNzS4xVuPnl070MuXJTnuUikU6c5ZliA3bioTx7aEcQbTw3afxWT_1MfogaU4CNrsK6pEkAB_JslnSzPp1MJxBt2qMvTLHTqe7NxDcQ_il1yKfRlj4dUNKQaGzXKweuauhO0a0MNvE4/s1600/picsym12.png" /></a></div>
The geometric series converges to <span class="mathquill-embedded-latex font16">\frac{5}{4}</span>.<br />
<br />
Next example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B5%7D%7B2%7D%5E%7Bn%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe4KMvssAUwF3VUV87XOgFAIaKjTiKwJ8mODBKc_7Eemrne8YWDWlpVRXGDtoMvdZMmNgAEi54-NZclJ26x8-LuOb-c58TutGPcXXTgvnxzFG5k2YOD8VeaV3D2n3_k37gPdtcaYp4N_s/s1600/picsym13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="45" data-original-width="111" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe4KMvssAUwF3VUV87XOgFAIaKjTiKwJ8mODBKc_7Eemrne8YWDWlpVRXGDtoMvdZMmNgAEi54-NZclJ26x8-LuOb-c58TutGPcXXTgvnxzFG5k2YOD8VeaV3D2n3_k37gPdtcaYp4N_s/s1600/picsym13.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Reference the geometric series convergence test<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPK5N_CVvaKiUCaPZd1cWY1fntuYupOazbimJCdH1lXLlWIwVAy4oE5evDwZKxeB_Zr8ukWQ7vjdAaCixGsuEVSFQhDDhLuWl1ouMIlnYWKKjpWZ3PUGAnA4znmVVdcxLbWI1Xdq4U3jE/s1600/picsym14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="81" data-original-width="377" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPK5N_CVvaKiUCaPZd1cWY1fntuYupOazbimJCdH1lXLlWIwVAy4oE5evDwZKxeB_Zr8ukWQ7vjdAaCixGsuEVSFQhDDhLuWl1ouMIlnYWKKjpWZ3PUGAnA4znmVVdcxLbWI1Xdq4U3jE/s1600/picsym14.png" /></a></div>
<br />
<span style="white-space: pre;"> 2. </span>Determine the value of r<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqMBnfCbMnEX4hVj4ilq7Wf308Bib6VMGSZOV8hHe2nhWCSEeWT-_RLCjFPGnCqGTb4YxezGPUj6QOrN_GimDRdveebtncjXUBCF6wAL4j8CTFcHv67JtK7j3u3DDT2wc3oceuiZy_EUA/s1600/picsym15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="50" data-original-width="78" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqMBnfCbMnEX4hVj4ilq7Wf308Bib6VMGSZOV8hHe2nhWCSEeWT-_RLCjFPGnCqGTb4YxezGPUj6QOrN_GimDRdveebtncjXUBCF6wAL4j8CTFcHv67JtK7j3u3DDT2wc3oceuiZy_EUA/s1600/picsym15.png" /></a></div>
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<span style="white-space: pre;"> 3. </span>Determine if the series converges or diverges<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR4_eH4yeXEt0yPa-vqBmoCjAF-Hz_kbf9IwFqW1jzVrFSN7-uxivWUoekHl3B04Ul8SrA9-wo-Z-xWhZ2x3vAX67PHbD6j2yYlXSCCP7dW0Zy-E-xJOM6elTzNGjc1ivJ6262_8dLIFw/s1600/picsym16.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="47" data-original-width="116" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR4_eH4yeXEt0yPa-vqBmoCjAF-Hz_kbf9IwFqW1jzVrFSN7-uxivWUoekHl3B04Ul8SrA9-wo-Z-xWhZ2x3vAX67PHbD6j2yYlXSCCP7dW0Zy-E-xJOM6elTzNGjc1ivJ6262_8dLIFw/s1600/picsym16.png" /></a></div>
The geometric series diverges.<br />
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Last example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cleft(-%5Cfrac%7B5%7D%7B6%7D%5Cright)%5E%7Bn%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoTF8U-18pZ_to8ctdil8wRUusAihw6cDAafYrJnM_6o2LElkvGw-lGvHV3b3MAgLLR_HW_pMhwUIygTJ7d2T1ZnNL2rYcVs6yyNjngYPXCWYl-chy81tOgqNrpbFq7wn9Ty7fN-0tGJQ/s1600/picsym17.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="55" data-original-width="135" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoTF8U-18pZ_to8ctdil8wRUusAihw6cDAafYrJnM_6o2LElkvGw-lGvHV3b3MAgLLR_HW_pMhwUIygTJ7d2T1ZnNL2rYcVs6yyNjngYPXCWYl-chy81tOgqNrpbFq7wn9Ty7fN-0tGJQ/s1600/picsym17.png" /></a></div>
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<br />
<span style="white-space: pre;"> 1. </span>Reference the geometric series convergence test<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj-J1t2L8hDwilAZv48IRnDKeOaBFOLLRn-rxltg3jLk5kL23IxoUb3LLkQtq_n8yu4Akajvlm77FZSkdzg8i4jGIPKWhZryDr4f9oA9stnF0TaBas1dBLqXWRM3vghG8O_INK9RKwBbY/s1600/picsym18.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="81" data-original-width="377" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj-J1t2L8hDwilAZv48IRnDKeOaBFOLLRn-rxltg3jLk5kL23IxoUb3LLkQtq_n8yu4Akajvlm77FZSkdzg8i4jGIPKWhZryDr4f9oA9stnF0TaBas1dBLqXWRM3vghG8O_INK9RKwBbY/s1600/picsym18.png" /></a></div>
<br />
<span style="white-space: pre;"> 2. </span>Determine the value of r<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiC1B_ZkSYO-R-4P4kshNDLBVBPZrM7KsZ5E5hN46uKTClWMKO5Tybh8ZHLQjIrl3KgZLTiP7i4bSy5kGUyZX6Xei1OQenisllowVq1fAGqylTR7eIY22uz1hNicGDPh0J4N8o_6AsN7G4/s1600/picsym19.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="43" data-original-width="105" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiC1B_ZkSYO-R-4P4kshNDLBVBPZrM7KsZ5E5hN46uKTClWMKO5Tybh8ZHLQjIrl3KgZLTiP7i4bSy5kGUyZX6Xei1OQenisllowVq1fAGqylTR7eIY22uz1hNicGDPh0J4N8o_6AsN7G4/s1600/picsym19.png" /></a></div>
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<span style="white-space: pre;"> 3. </span>Determine if the series converges or diverges<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4XwkxqDqDrPODE1BNxKyU-GtRP2MJN0eUHWFwKDxp8sLwKsAZzBuHm1CTTHdbY4EkvI26nheZadSJ6jpSJJEDkBeHz3aTLHb78EjQ-RUYjm43ZZQDRJ9b0wOcMXYIue-Ev7ElfpHyTSY/s1600/picsym20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="47" data-original-width="118" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4XwkxqDqDrPODE1BNxKyU-GtRP2MJN0eUHWFwKDxp8sLwKsAZzBuHm1CTTHdbY4EkvI26nheZadSJ6jpSJJEDkBeHz3aTLHb78EjQ-RUYjm43ZZQDRJ9b0wOcMXYIue-Ev7ElfpHyTSY/s1600/picsym20.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsTRVL_wkhbhzR_VYVkN-C2ohdIdHekMeKpiDcGUDFoqB18QoUIt9NQaiUWVKTMh2-L-RoWROCGN46DBjCn9ywO5R8vKw1W8NzvJ1SRQzx41MpiR4OyaR0_f1ZYL_2bfKK6BATMjN8tpg/s1600/picsym21.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="107" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsTRVL_wkhbhzR_VYVkN-C2ohdIdHekMeKpiDcGUDFoqB18QoUIt9NQaiUWVKTMh2-L-RoWROCGN46DBjCn9ywO5R8vKw1W8NzvJ1SRQzx41MpiR4OyaR0_f1ZYL_2bfKK6BATMjN8tpg/s1600/picsym21.png" /></a></div>
The geometric series converges to 6.<br />
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As you can see, it is not too difficult to determine if a geometric series converges or not. After doing some practice problems, you’ll get the hang of it very quickly. For more help or practice on geometric series, check out Symbolab’s <span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice/series-practice?subTopic=Geometric%20Series%20Test"><span style="color: #1155cc;">Practice</span></a></span>. Next blog post, I’ll go over the convergence test for p-series.<br />
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Until next time,<br />
<br />
LeahUnknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-60103356904246201722019-03-25T10:01:00.001-07:002022-06-08T07:56:46.138-07:00Advanced Math Solutions - Matrix Rank Calculator, MatricesIn the last two blog posts, we talked about Row Echelon Form (REF) and Reduced Row Echelon Form (RREF). In this blog post, we will talk about matrix rank. Determining a matrix’s rank will involve using REF or RREF, so make sure to review those blog posts before continuing on.<br />
<br />
The rank of matrix is the dimension of the vector space created by its columns or rows. It is important to note that column rank and row rank are the same thing. We will find the rank of the matrix, by using the row rank.<br />
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Another way to think of this is that the rank of a matrix is the number of linearly independent rows or columns. Linearly independent means that no rows or columns can be the combination of the other rows or columns.<br />
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For example:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW9BbhQEOTVZvwmV36zxCaMmjoNMJ0TbFpYPYVKYXaKM74itL3vaVh9wWJSOPdJozNDIlLXoJ6rlc654fd7H-Arkrjl3HnRsb5-zNOh04FR2Swv_1PWnst0Tv7C6VqZ4rnaTwwhHKQgqs/s1600/picsym1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="85" data-original-width="143" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW9BbhQEOTVZvwmV36zxCaMmjoNMJ0TbFpYPYVKYXaKM74itL3vaVh9wWJSOPdJozNDIlLXoJ6rlc654fd7H-Arkrjl3HnRsb5-zNOh04FR2Swv_1PWnst0Tv7C6VqZ4rnaTwwhHKQgqs/s1600/picsym1.png" /></a></div>
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Here, Row 2 is a combination of Row 1 and Row 3 (Row 1 + Row 3). Therefore the rows are not linearly independent.<br />
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In order to determine the rank of a matrix:<br />
<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Put the matrix in REF or RREF</blockquote>
<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Count the number of non-zero rows </blockquote>
Let’s see some examples. Please note that I won’t be going over how to put the matrices in REF or RREF.<br />
<br />
First example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/rank%20%5Cbegin%7Bpmatrix%7D1%20%26%202%20%26%204%20%5C%5C%208%20%26%20-5%20%26%2010%20%5C%5C%200%26%202%26%201%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxykq3e347N_v3zTN8GtcACExt02K-tL1zCKw1gw06a41D_m6-27McOjTyq82ENR2k7rjjOSNZwohQzGxv6yECK8EgRT8j-4ymo2KizVhiqfobOB7OeeHYTwDVPalbnTzYiuoUxRjvwow/s1600/picsym2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="87" data-original-width="226" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxykq3e347N_v3zTN8GtcACExt02K-tL1zCKw1gw06a41D_m6-27McOjTyq82ENR2k7rjjOSNZwohQzGxv6yECK8EgRT8j-4ymo2KizVhiqfobOB7OeeHYTwDVPalbnTzYiuoUxRjvwow/s1600/picsym2.png" /></a></div>
<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Put the matrix in REF or RREF</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQmBCwdTeB-Pv_G3X2D9zsvfD2mS3a767hGA98rn3wHrYTVC2viVi5Sig7SRBgRFUs0mzQMVaQoY8xDLc6M5F4AN7ZjJT70DO40Cj_pNgeLkD63DlN1ugZq2E2KE-noWzQthdDaKObvgM/s1600/picsym3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="86" data-original-width="122" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQmBCwdTeB-Pv_G3X2D9zsvfD2mS3a767hGA98rn3wHrYTVC2viVi5Sig7SRBgRFUs0mzQMVaQoY8xDLc6M5F4AN7ZjJT70DO40Cj_pNgeLkD63DlN1ugZq2E2KE-noWzQthdDaKObvgM/s1600/picsym3.png" /></a></div>
The matrix is in RREF.<br />
<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Count the number of non-zero rows</blockquote>
There are 3 non-zero rows. The rank of this matrix is 3.<br />
<br />
Not so bad! Next example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/rank%20%5Cbegin%7Bpmatrix%7D2%264%266%5C%5C%20%20%20%20%201%260%262%5C%5C%20%20%20%20%20-2%266%261%5C%5C%20%20%20%20%208%26-1%2610%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1qgBlq6yk4tWD-bVU5CJ79fy7RS9PMRvkkO6nL7dr1iJhtSNDcBbUQeXPrmKyWcYIaWTP2n82vQgMrAJUdeCDQplYyTOpyLorxMnZyyyirJ_kasgDaPSHDkBzsnk1DnD-Zv37qvBLZWc/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="104" data-original-width="186" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1qgBlq6yk4tWD-bVU5CJ79fy7RS9PMRvkkO6nL7dr1iJhtSNDcBbUQeXPrmKyWcYIaWTP2n82vQgMrAJUdeCDQplYyTOpyLorxMnZyyyirJ_kasgDaPSHDkBzsnk1DnD-Zv37qvBLZWc/s1600/picsym4.png" /></a></div>
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<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Put the matrix in REF or RREF</blockquote>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2Hz8I5DgUOWnpyYdDpFdsvzImT9RiX-9uSXI-wdR5I1BD-I0Fy84CTV60OKsZrhiNT-WGeaTqm0vhOAryq5NGJk11cSI9Mh3S9e5YYNulAZDqnv5UEtxXqQONr9Jt-b_YmsnDqiSkEM4/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="144" data-original-width="138" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2Hz8I5DgUOWnpyYdDpFdsvzImT9RiX-9uSXI-wdR5I1BD-I0Fy84CTV60OKsZrhiNT-WGeaTqm0vhOAryq5NGJk11cSI9Mh3S9e5YYNulAZDqnv5UEtxXqQONr9Jt-b_YmsnDqiSkEM4/s1600/picsym5.png" /></a></div>
The matrix is in REF.<br />
<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Count the number of non-zero rows</blockquote>
There are 3 non-zero rows. The rank of this matrix is 3.<br />
<br />
Last example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/rank%5Cbegin%7Bpmatrix%7D1%261%260%262%5C%5C%201%261%261%260%5C%5C%200%26-1%261%261%5C%5C%20-1%260%26-1%260%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuBXCVB_QK1XSzEH2lXe-tTsIG1hvvMHb7ET5LX573GMjqGDQhU3X-ncHlvzcM2bvJK532KCgjIJET5qAJdrbn-Dn717OHkV2oiXiWOnzJH_5fMjSFtkSt0h1mVjFF6cuHRccka22pEQs/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="107" data-original-width="251" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuBXCVB_QK1XSzEH2lXe-tTsIG1hvvMHb7ET5LX573GMjqGDQhU3X-ncHlvzcM2bvJK532KCgjIJET5qAJdrbn-Dn717OHkV2oiXiWOnzJH_5fMjSFtkSt0h1mVjFF6cuHRccka22pEQs/s1600/picsym6.png" /></a></div>
<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Put the matrix in REF or RREF</blockquote>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpAWWfcKuGd-SKfFGO73l57SUn91piN_2TuYvFvPflaExMRHNCXL1ikiP1m1uConV8VXfnpUO2r7F9NvMe7ELaK_9g590UmP4nuNsgBtXQQyGb7k2Y79LieOvbtina0PUeutf0n-pesjA/s1600/picsym7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="104" data-original-width="137" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpAWWfcKuGd-SKfFGO73l57SUn91piN_2TuYvFvPflaExMRHNCXL1ikiP1m1uConV8VXfnpUO2r7F9NvMe7ELaK_9g590UmP4nuNsgBtXQQyGb7k2Y79LieOvbtina0PUeutf0n-pesjA/s1600/picsym7.png" /></a></div>
The matrix is in RREF.<br />
<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Count the number of non-zero rows</blockquote>
There are 4 non-zero rows. The rank of this matrix is 4.<br />
<br />
As you can see, finding the rank of a matrix is not hard. You just have to make sure you’ve mastered putting matrices in REF and RREF.<br />
<br />
For more help or practice on this topic, check out Symbolab’s <span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice"><span style="color: #1155cc;">Practice</span></a></span>.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-38334044737304606472019-03-03T12:11:00.001-08:002022-06-08T07:56:46.887-07:00High School Math Solutions - Matrix Inverse Calculator, Matrices (Part 2)<div>
In the last two blog posts, I talked about how to find the <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="http://blog.symbolab.com/2018/12/advanced-math-solutions-matrix-inverse.html"><span style="color: #1155cc;">inverse </span></a></span></u>of a matrix and how to calculate the <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="http://blog.symbolab.com/2019/01/advanced-math-solutions-matrix-inverse.html"><span style="color: #1155cc;">determinant </span></a></span></u>of the matrix. Please review these two blog posts before continuing, if you are not familiar with either topic.</div>
<div>
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<div>
As you saw in the inverse blog post, calculating the inverse of a matrix can require a lot of steps and some time. In this blog post, I will go over a shortcut for calculating the inverse of a 2x2 matrix.</div>
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Here are the steps for calculating the inverse of a 2x2 matrix, using the shortcut:</div>
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<div>
<span style="white-space: pre;"> 1. </span>Calculate the determinant of matrix A<br />
Reminder:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHVuCj8nedfmxAAnyod5GcdG3dmEy6e4_LyAxyBbBmIWwUqmdJlGL0rXWmDXFF84p6LumOfD42cnQqLMPEv51R4qDrtgMSoqWgFAv91rzhXWjlD5OcYsIjMX9qCR46psE57Lww5q-50fs/s1600/picsym1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="63" data-original-width="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHVuCj8nedfmxAAnyod5GcdG3dmEy6e4_LyAxyBbBmIWwUqmdJlGL0rXWmDXFF84p6LumOfD42cnQqLMPEv51R4qDrtgMSoqWgFAv91rzhXWjlD5OcYsIjMX9qCR46psE57Lww5q-50fs/s1600/picsym1.png" /></a></div>
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<div>
<span style="white-space: pre;"> 2. </span>Reorganize matrix A<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMWcb6KV0fc2lR-t_Rs8Fj4lUeZg251O-FgBTK8ZLHicHNMeWimu4hLqoheM8b2arDvf_j2XqRR-xRVoWAL1z8vU3A339PQ2SVpJnM8k1h4UNkdqwPOILgshPIS-O40TGusISlb3wF_2E/s1600/picsym2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMWcb6KV0fc2lR-t_Rs8Fj4lUeZg251O-FgBTK8ZLHicHNMeWimu4hLqoheM8b2arDvf_j2XqRR-xRVoWAL1z8vU3A339PQ2SVpJnM8k1h4UNkdqwPOILgshPIS-O40TGusISlb3wF_2E/s1600/picsym2.png" /></a></div>
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<div>
<span style="white-space: pre;"> 3. </span>Multiply matrix A by <span class="mathquill-embedded-latex font16">\frac{1}{det(A)}</span></div>
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<div>
These steps can be summarized by this formula:<br />
<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBRKBdWZA-1L7caDbrzIgxsCLxVaf4k088_3Zd8ZUWn-bBFzboGoud-7knqm6rcQ6SXBQd9bYZka6s7SvkAyaEYZv22YemNMkN3l637f6T2kWlwIQx5hQlm8a5Th07KP3FzzM9UTZfMho/s1600/picsym3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="70" data-original-width="293" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBRKBdWZA-1L7caDbrzIgxsCLxVaf4k088_3Zd8ZUWn-bBFzboGoud-7knqm6rcQ6SXBQd9bYZka6s7SvkAyaEYZv22YemNMkN3l637f6T2kWlwIQx5hQlm8a5Th07KP3FzzM9UTZfMho/s1600/picsym3.png" /></a></div>
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Not too difficult, right? Let’s see some examples.</div>
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First example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D2%26%204%20%5C%5C%201%26%206%5Cend%7Bpmatrix%7D%5E%7B-1%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVxeQdWhwJnqkSj3-fwk1YvcX4JsmyJ9dD4DzAtvpH36o1ngseJ6kf4b_QGQzKAoSf5FCGdsyi7_k7n1SQ_ddWKVwJbbKYbVVzj3qLqKoK-6Hw5xWowDsVRjm2UZpryrjPfd28cl-bcvA/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="113" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVxeQdWhwJnqkSj3-fwk1YvcX4JsmyJ9dD4DzAtvpH36o1ngseJ6kf4b_QGQzKAoSf5FCGdsyi7_k7n1SQ_ddWKVwJbbKYbVVzj3qLqKoK-6Hw5xWowDsVRjm2UZpryrjPfd28cl-bcvA/s1600/picsym4.png" /></a></div>
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<br /></div>
</div>
<div>
<span style="white-space: pre;"> 1. </span>Calculate the determinant of the matrix<br />
<br /></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg25x_VvKhhuhqYm4Lok__W6q9MBCGTOUVSjBJSc_RAwq0S2Btm-S18bYDP7A9GYVKNsJcrfgVNgJW5X9LiTcGb7vwDJTqN7uo9EoAV_RKMbiyE-C7hygLO07QbpenYTE-MdEuAbQMs8sY/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="63" data-original-width="237" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg25x_VvKhhuhqYm4Lok__W6q9MBCGTOUVSjBJSc_RAwq0S2Btm-S18bYDP7A9GYVKNsJcrfgVNgJW5X9LiTcGb7vwDJTqN7uo9EoAV_RKMbiyE-C7hygLO07QbpenYTE-MdEuAbQMs8sY/s1600/picsym5.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqvH5LYDeoCpmrd13lwtBSWzd3HC6X8xt_BHy9wpsIiCh0_LFnYn968rju9WwNKu21XVkfCs1Z0nPgmLv8iGgxcj-W2g9PhkrP8AhL2_spMhtEFX4T5BSn3Wx9qWrGYUWNypFzrFh8z-Q/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="66" data-original-width="139" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqvH5LYDeoCpmrd13lwtBSWzd3HC6X8xt_BHy9wpsIiCh0_LFnYn968rju9WwNKu21XVkfCs1Z0nPgmLv8iGgxcj-W2g9PhkrP8AhL2_spMhtEFX4T5BSn3Wx9qWrGYUWNypFzrFh8z-Q/s1600/picsym6.png" /></a></div>
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<div>
<span style="white-space: pre;"> 2. </span>Reorganize the matrix<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ATyIZQu3pa3wloDEuX7-o_rjwxYAqxIpKBohFz96TKxFKqlZumB9cN4jNs1rJAu14xII6Ja7xC9avR5QTq9k2j-0o1ye0f3ZYgx-opqycaXQtVxjWJ-x4FbQjH8f1EAr4HZAs7wfwGg/s1600/picsym7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ATyIZQu3pa3wloDEuX7-o_rjwxYAqxIpKBohFz96TKxFKqlZumB9cN4jNs1rJAu14xII6Ja7xC9avR5QTq9k2j-0o1ye0f3ZYgx-opqycaXQtVxjWJ-x4FbQjH8f1EAr4HZAs7wfwGg/s1600/picsym7.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibG4vchzQ-clNpxfzk4PUXMdQM9gmqfcTrMipw64m9LYYke92NMGrLPWWKoknbVUBswecKp1pYr8u1L7PtJlfWkrWc4ujy1e7vczwyrqlG4TBMbs9b1GBtwPn2mnJJ9yk-LQITjcNEU6Y/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="67" data-original-width="108" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibG4vchzQ-clNpxfzk4PUXMdQM9gmqfcTrMipw64m9LYYke92NMGrLPWWKoknbVUBswecKp1pYr8u1L7PtJlfWkrWc4ujy1e7vczwyrqlG4TBMbs9b1GBtwPn2mnJJ9yk-LQITjcNEU6Y/s1600/picsym8.png" /></a></div>
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<div>
<span style="white-space: pre;"> 3. </span>Multiple the matrix by <span class="mathquill-embedded-latex font16">\frac{1}{det(A)}</span><br />
<span class="mathquill-embedded-latex font16"><br /></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV-UPJjQnJKnLPvMPS33uiiUvb0884tNqSECS2_56PHDhF6Ce9g97_FsippKanW0uulwUBD0zp_qnX9VACYsXUOoopKmVy57r0wybytzrXDoBQNxUZVWT-vxQiY2n2rDZECCg39fLL8ZY/s1600/picsym9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="155" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV-UPJjQnJKnLPvMPS33uiiUvb0884tNqSECS2_56PHDhF6Ce9g97_FsippKanW0uulwUBD0zp_qnX9VACYsXUOoopKmVy57r0wybytzrXDoBQNxUZVWT-vxQiY2n2rDZECCg39fLL8ZY/s1600/picsym9.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihPKGwBdDY0P2RFxTrNzP6KJ_yujDItIN3_ebh1J1sYfxBoDjurigY-Mj1oo0GZqe30npCeqLiCO67xcd5hiTY9iUacGi1taDi9ql2pnZUGLZY113fjJIxAKPmUe6rCoZWEehGRuLBsdo/s1600/picsym10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="104" data-original-width="149" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihPKGwBdDY0P2RFxTrNzP6KJ_yujDItIN3_ebh1J1sYfxBoDjurigY-Mj1oo0GZqe30npCeqLiCO67xcd5hiTY9iUacGi1taDi9ql2pnZUGLZY113fjJIxAKPmUe6rCoZWEehGRuLBsdo/s1600/picsym10.png" /></a></div>
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Next example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/inverse%20%5Cbegin%7Bpmatrix%7D-3%262%5C%5C%20%20%205%2610%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrJy1RkQBTKmFmUqqS4LWq-iiEqqIQV8mWKJQqsC6NVx1uOxE1FOV3i9HSNqe4IgZ6iJyqwyFq3BNK0lQAOdPYg7FOvaWS7uGuS7scJb7iEZZ9WHTTCqh_rBsVs36q-q_Z8SD6mUBzwGw/s1600/picsym12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="57" data-original-width="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrJy1RkQBTKmFmUqqS4LWq-iiEqqIQV8mWKJQqsC6NVx1uOxE1FOV3i9HSNqe4IgZ6iJyqwyFq3BNK0lQAOdPYg7FOvaWS7uGuS7scJb7iEZZ9WHTTCqh_rBsVs36q-q_Z8SD6mUBzwGw/s1600/picsym12.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Calculate the determinant of the matrix<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWKeVZ-pJjWUMs0O20d7ylVmWwfId7gDfOfniYZee2D3QdcqvN_6A-nTD6wtEOal7SxZ-WBReFL_7QjkdGb13aE911V3KWGEqkPVfTU-gbJVCI-It1FYiQopeN5foJ1uYnDuHv2cfmMis/s1600/picsym13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="277" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWKeVZ-pJjWUMs0O20d7ylVmWwfId7gDfOfniYZee2D3QdcqvN_6A-nTD6wtEOal7SxZ-WBReFL_7QjkdGb13aE911V3KWGEqkPVfTU-gbJVCI-It1FYiQopeN5foJ1uYnDuHv2cfmMis/s1600/picsym13.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Reorganize the matrix </div>
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<span style="white-space: pre;"> 3. </span>Multiple the matrix by <span class="mathquill-embedded-latex font16">\frac{1}{det(A)}</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDhZAC7HfL6Q45fRt4mmf1nlVrCURVhZA_jq4jtyzpN76CynR3vw91rQ1kfwU0m1Ahs5rvAZF_0azYbkUQF9Sjv0NMuZSsM7qptywd1GdsnELmGMjSiQksKSJbztnytA0Z2s7HPMqVBrY/s1600/picsym17.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="60" data-original-width="172" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDhZAC7HfL6Q45fRt4mmf1nlVrCURVhZA_jq4jtyzpN76CynR3vw91rQ1kfwU0m1Ahs5rvAZF_0azYbkUQF9Sjv0NMuZSsM7qptywd1GdsnELmGMjSiQksKSJbztnytA0Z2s7HPMqVBrY/s1600/picsym17.png" /></a></div>
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Last example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/inverse%5Cbegin%7Bpmatrix%7D12%265%5C%5C%2011%266%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfDfJMSQZ1lc7j3vJTPoHSxXKtjeQBtQ975s78zk2C0HyUU6bcKMT4X-F53Hq0Xbfvy37TxLSjToouhjZ52b9eTQg-aajnzTKMq50dRxeAoO_RnCjL3n0Y785Ch82nrJAEkfSAzPOecwY/s1600/picsym19.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="148" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfDfJMSQZ1lc7j3vJTPoHSxXKtjeQBtQ975s78zk2C0HyUU6bcKMT4X-F53Hq0Xbfvy37TxLSjToouhjZ52b9eTQg-aajnzTKMq50dRxeAoO_RnCjL3n0Y785Ch82nrJAEkfSAzPOecwY/s1600/picsym19.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Calculate the determinant of the matrix<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikHvZSzYZil5sD0SiIENOfOWj2wVZk2qR0fYNgX1ce0afecdxRP7DfLN1Gg1YHqyDhMUJsn_ldW__nbgngVbyZ5ynwUmGvEJi7TYHMVFzS-JRNLhhA8Qrg6GeNOmVWwqO4ovUXzSTSZNk/s1600/picsym20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="58" data-original-width="258" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikHvZSzYZil5sD0SiIENOfOWj2wVZk2qR0fYNgX1ce0afecdxRP7DfLN1Gg1YHqyDhMUJsn_ldW__nbgngVbyZ5ynwUmGvEJi7TYHMVFzS-JRNLhhA8Qrg6GeNOmVWwqO4ovUXzSTSZNk/s1600/picsym20.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Reorganize the matrix </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgztFreEVNN9uTo0nI4ClkOI89xIQOdPVh3I4A5AKuoPO4GpX3_7itmpimVpmobJcpYbRVKNpllWMisiRFu-K7hpfNC6BO-j9g0-4Wba64LYbP3YiJ1iNw2h6vPRKfL-SotVt_ffuFoYB0/s1600/picsym22.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgztFreEVNN9uTo0nI4ClkOI89xIQOdPVh3I4A5AKuoPO4GpX3_7itmpimVpmobJcpYbRVKNpllWMisiRFu-K7hpfNC6BO-j9g0-4Wba64LYbP3YiJ1iNw2h6vPRKfL-SotVt_ffuFoYB0/s1600/picsym22.png" /></a></div>
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<span style="white-space: pre;"> 3. </span>Multiple the matrix by <span class="mathquill-embedded-latex font16">\frac{1}{det(A)}</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoRSrNt43OtsCdCh_DPAcfIMtrBrEfWlb1z1i4OCl4p23qHLSqizycAkAP86SVh3tbZhZG2lcoEwKLSwkNN1hJRfls3d1KtF2c4GNSbksN7aTLRNG7CsU8UBF5IZq2qKFoA04JoD4yA_U/s1600/picsym24.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="65" data-original-width="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoRSrNt43OtsCdCh_DPAcfIMtrBrEfWlb1z1i4OCl4p23qHLSqizycAkAP86SVh3tbZhZG2lcoEwKLSwkNN1hJRfls3d1KtF2c4GNSbksN7aTLRNG7CsU8UBF5IZq2qKFoA04JoD4yA_U/s1600/picsym24.png" /></a></div>
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This shortcut will help make calculating the inverse of a 2x2 matrix easier. That concludes our blog post series on matrices! For more help or practice on this topic, check out Symbolab’s <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice/matrices-practice?subTopic=Determinant"><span style="color: #1155cc;">Practice</span></a></span></u>.</div>
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Until next time,</div>
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Leah</div>
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Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-25753992210842997462019-01-08T11:58:00.001-08:002022-06-08T07:56:47.640-07:00Advanced Math Solutions - Matrix Inverse Calculator, Determinants<a href="https://blog.symbolab.com/2018/12/advanced-math-solutions-matrix-inverse.html" target="_blank">Last blog</a> post, I talked about what the inverse of a matrix is. In this blog post, I will go over what the determinant of matrix is and how to calculate.<br />
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The <b>determinant</b> is a value calculated from an <span class="mathquill-embedded-latex font16">n\times n</span> matrix. The determinant of a matrix, A, can be denoted as det(A), det A, or |A|. There are many uses for determinants. The determinant can be used to solve a system of equations. The determinant can tell you if the matrix is invertible or not (it is not if the matrix is 0).<br />
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We will discuss three ways to solve three different types of matrices.<br />
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<b>Determinant of a 2x2 matrix:</b><br />
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<b>Determinant of a 3x3 matrix:</b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7AUqyogdtdQzdicwWfAIT7K1HUzu-mWB952azFmyOiiSVecZuAru2Wh99147prSy2gwOQz2b3T82pUSDP-qc6SMgG7zFuOE86wOswHF7zgTa-JyGGQFbo03gWb7s-PwLZdqs7k9MfEHs/s1600/picsym2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><br />
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<b>Determinant of a nxn matrix that is 4x4 or larger:</b><br />
<br />
<span style="white-space: pre;"> 1. </span>Put the matrix in REF (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="http://blog.symbolab.com/2018/10/advanced-math-solutions-matrix-row.html"><span style="color: #1155cc;">here</span></a></span></u> is the blog post on REF for reference)<br />
<blockquote class="tr_bq">
<ul>
<li>Make note of how many times you swapped rows to achieve putting the matrix in REF </li>
</ul>
</blockquote>
<span style="white-space: pre;"> 2. </span>Calculate the product of the elements in the diagonal<br />
<blockquote class="tr_bq">
<ul>
<li>If no rows were swapped, this is the determinant</li>
</ul>
</blockquote>
<div style="text-align: left;">
<span style="white-space: pre;"> 3. </span>If rows were swapped, multiply the product of the elements in the diagonal by (-1) raised to the number of times rows were swapped. This will give you the determinant.</div>
<blockquote class="tr_bq">
<ul>
<li>Swapping rows changes the sign of the determinant</li>
</ul>
</blockquote>
<br />
Let’s see some examples to better understand how to calculate the determinant.<br />
<br />
First example (click <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cdet%5Cbegin%7Bpmatrix%7D-1%26%202%20%5C%5C%20%201%20%26%204%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzEjOBMgQrQNgKvTu91deiuOhAzwSTueYwWEAgH-YgRbsBo-upBu_WEUjjQZLKLEPLRK82eD1NvbYI4uO-J-bJH6nVf8gyzCp5Cflkkk2mgOXF9NLCo0CTDW3cjOgAS6Zfc8ClIo6IkNw/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="140" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzEjOBMgQrQNgKvTu91deiuOhAzwSTueYwWEAgH-YgRbsBo-upBu_WEUjjQZLKLEPLRK82eD1NvbYI4uO-J-bJH6nVf8gyzCp5Cflkkk2mgOXF9NLCo0CTDW3cjOgAS6Zfc8ClIo6IkNw/s1600/picsym4.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Use this formula:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9Af5UJvYBZeZa9jzize1v2kt4FTAcPw69tfhYkTiDWH1cCrb3wWR4G3gKw1UigXl2zn4iDPsMWFQVirn1-OjttkmjmNi_UksWAyx-wALkz4MdtCXy9rcGBbINHGLSvkQE2B0FVJ3L_pc/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="43" data-original-width="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9Af5UJvYBZeZa9jzize1v2kt4FTAcPw69tfhYkTiDWH1cCrb3wWR4G3gKw1UigXl2zn4iDPsMWFQVirn1-OjttkmjmNi_UksWAyx-wALkz4MdtCXy9rcGBbINHGLSvkQE2B0FVJ3L_pc/s1600/picsym5.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcO3y5DZOVL_4FThS3QQIilJSoDLkvwEMUIp_8EPZ8-zZZDDI_PJp1bPW_QrA8dpbFzuFw6xZ8jRStwVhDxfeh68J8X-173k17wj-OmZ6YNYzJfW0NfgAmrOJ8nk8sPLchHNakk_EmdHQ/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="34" data-original-width="161" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcO3y5DZOVL_4FThS3QQIilJSoDLkvwEMUIp_8EPZ8-zZZDDI_PJp1bPW_QrA8dpbFzuFw6xZ8jRStwVhDxfeh68J8X-173k17wj-OmZ6YNYzJfW0NfgAmrOJ8nk8sPLchHNakk_EmdHQ/s1600/picsym6.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigHu5bhbTcTzGiCsM_DOCToAdrz6Vk17y1rAgd9YB7d22RuCE-Yhbomncn46d7vG3HPTiiNp24e4CKEZ70TwWJef85z0kOy6vsoRY82_mfuoNjbYKqUZd0rHunoDhIsZ2kd4C6xp_l5Lg/s1600/picsym7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="30" data-original-width="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigHu5bhbTcTzGiCsM_DOCToAdrz6Vk17y1rAgd9YB7d22RuCE-Yhbomncn46d7vG3HPTiiNp24e4CKEZ70TwWJef85z0kOy6vsoRY82_mfuoNjbYKqUZd0rHunoDhIsZ2kd4C6xp_l5Lg/s1600/picsym7.png" /></a></div>
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Next example (click <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cdet%5Cbegin%7Bpmatrix%7D1%20%26%202%20%26-2%20%5C%5C%204%20%26%201%266%20%5C%5C%204%26%203%20%260%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">here</span></a></span></u>):<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOPGGpX0JTBjzVI77HiFYg_gXHkqihyJDzMnLmVuGvucETbnLsWkVOJkbVoWUXEVqzV9Q7y0tH-graOe2b5BkaYPsfsUgB4OhdgtFs8KEuOiNFqPKLAXfi2_DUw2fF3iPdMq0l2XAvY30/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="87" data-original-width="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOPGGpX0JTBjzVI77HiFYg_gXHkqihyJDzMnLmVuGvucETbnLsWkVOJkbVoWUXEVqzV9Q7y0tH-graOe2b5BkaYPsfsUgB4OhdgtFs8KEuOiNFqPKLAXfi2_DUw2fF3iPdMq0l2XAvY30/s1600/picsym8.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Use this formula: <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsLKyb4V0cRKj65HrrUlS2hmKUJU2Tzh1zWLdtqB2O99A5MDWLqD88VbDZ0XGSFo6-qZGmCEnlWuxmBG-xMyVYHAOMPnMqUWxvmXJ6lzhRKocSdQUxhNcgJ1XYKIt6zmnrawW8J8Dr_nA/s1600/picsym9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="65" data-original-width="432" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsLKyb4V0cRKj65HrrUlS2hmKUJU2Tzh1zWLdtqB2O99A5MDWLqD88VbDZ0XGSFo6-qZGmCEnlWuxmBG-xMyVYHAOMPnMqUWxvmXJ6lzhRKocSdQUxhNcgJ1XYKIt6zmnrawW8J8Dr_nA/s1600/picsym9.png" /></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjEZ5GwcdN5f4DA4x-BB_5kzr5YEsaAOKCK9YLSWY806Kf3nu09EzBSA3YAeCBUtBMVZjAedxHXXLXyhxRfikPxeDxRUMVwtOca9d-gDu0UeCuiTqPJp7ZLw5he5HSd0VL-htsJvzSOVQ/s1600/picsym10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="393" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjEZ5GwcdN5f4DA4x-BB_5kzr5YEsaAOKCK9YLSWY806Kf3nu09EzBSA3YAeCBUtBMVZjAedxHXXLXyhxRfikPxeDxRUMVwtOca9d-gDu0UeCuiTqPJp7ZLw5he5HSd0VL-htsJvzSOVQ/s1600/picsym10.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn8rh02bUcfXlMR0Vpq3jD5pQz537dsLPKWSUTwjw_Q9XWWVjzXB4KRuDpX_ewPnOkJEHitMZdd6aIAVH1tQipF-bz1l9oCDdp4AMgnKS0MbKsrS9g_nSlAMTZk1H3Ok89WJMxn3_PZuw/s1600/picsym11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="28" data-original-width="487" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn8rh02bUcfXlMR0Vpq3jD5pQz537dsLPKWSUTwjw_Q9XWWVjzXB4KRuDpX_ewPnOkJEHitMZdd6aIAVH1tQipF-bz1l9oCDdp4AMgnKS0MbKsrS9g_nSlAMTZk1H3Ok89WJMxn3_PZuw/s1600/picsym11.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibVX3ju2IJE86Mm6tdvXwxw2j5ihEu8JaNNzYmfBunYZUWPv1qDPZ6FybmSCqQt1KPmFQZay2y2df9iCqCi7YSHeArPGUnpMYBAiuMQ4f-q-aMVp0weLmMhCADcmdLX2XIWRQmuKsBC00/s1600/picsym12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="31" data-original-width="287" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibVX3ju2IJE86Mm6tdvXwxw2j5ihEu8JaNNzYmfBunYZUWPv1qDPZ6FybmSCqQt1KPmFQZay2y2df9iCqCi7YSHeArPGUnpMYBAiuMQ4f-q-aMVp0weLmMhCADcmdLX2XIWRQmuKsBC00/s1600/picsym12.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp-aptfjX2BqbnpxlgP3KIAqqzBltT1QFPPvSQ4Op_5ni9XABFvdRJkLnIKvE2lkYrq-Lx4lmjn2GgbEj-aysLQrOHd4GnX8AQflE-uR3tXJtrcAeJacNzu5eRmKsRcqlARyk9N5wI08I/s1600/picsym13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="30" data-original-width="165" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp-aptfjX2BqbnpxlgP3KIAqqzBltT1QFPPvSQ4Op_5ni9XABFvdRJkLnIKvE2lkYrq-Lx4lmjn2GgbEj-aysLQrOHd4GnX8AQflE-uR3tXJtrcAeJacNzu5eRmKsRcqlARyk9N5wI08I/s1600/picsym13.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOko3yTd-UdF-69gOPvNEk5pV95AvAN__QprouKyOwSywKo4kEn17nlf_DGueTnsvU2SljJ3w3BzF1Vlp3Jzze0gzS56X7F0ivGoCTqH6CsL24edxRxQfd2ajs6MtPXHSqF2XtdqRnsno/s1600/picsym14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="28" data-original-width="48" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOko3yTd-UdF-69gOPvNEk5pV95AvAN__QprouKyOwSywKo4kEn17nlf_DGueTnsvU2SljJ3w3BzF1Vlp3Jzze0gzS56X7F0ivGoCTqH6CsL24edxRxQfd2ajs6MtPXHSqF2XtdqRnsno/s1600/picsym14.png" /></a></div>
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Last example (click <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cdet%5Cbegin%7Bpmatrix%7D1%26%202%20%26%204%20%26%201%20%5C%5C%20%201%20%26%200%20%26%202%26%206%20%5C%5C%20%204%20%261%261%264%5C%5C%204%26%202%20%26%200%20%262%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHO5WhyphenhyphenBVCm0LqEXwPebDFrO4v3I_HTXTfO2riKdBpp9EAFTZXd2-lUzOLrxQwpTi6wC6QL1BK2AePHHUmoGvN9NeEZ9cTJaHVFIlo5I6-qddnGGW7ayzLfejxbUrUeBBwVszb4wEBKrc/s1600/picsym15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="109" data-original-width="212" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHO5WhyphenhyphenBVCm0LqEXwPebDFrO4v3I_HTXTfO2riKdBpp9EAFTZXd2-lUzOLrxQwpTi6wC6QL1BK2AePHHUmoGvN9NeEZ9cTJaHVFIlo5I6-qddnGGW7ayzLfejxbUrUeBBwVszb4wEBKrc/s1600/picsym15.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Put the matrix in REF<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaAxqFkI7M_6tWnt-2I47opAou26oGu1UptC-s-ja6CY9CippHRc1L3_R-OhuKQg2uYR6KB8WT1Joa7Q1NmpLl0B3BcbWLvxQOMkJgbOHB7ce9V7HMiJoTHFfhWLSef714K1KO0t0A6zI/s1600/picsym16.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="163" data-original-width="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaAxqFkI7M_6tWnt-2I47opAou26oGu1UptC-s-ja6CY9CippHRc1L3_R-OhuKQg2uYR6KB8WT1Joa7Q1NmpLl0B3BcbWLvxQOMkJgbOHB7ce9V7HMiJoTHFfhWLSef714K1KO0t0A6zI/s1600/picsym16.png" /></a></div>
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Since there is already a blog post on how to put a matrix in REF, I am not going to go through the steps for doing this. You can look at the link for this last example to see how to do this.<br />
<br />
Note that rows were swapped 3 times to achieve putting the matrix in REF.<br />
<br />
<span style="white-space: pre;"> 2. </span>Calculate the product of the diagonal<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvjoBBtDkJpjYhgP6_zGKg3d3PfYn1w1Milsg-U2AEgoOQv1wvs7SAgraVcPOfy4ShU0LJlLt1kci7JJwWradY5lOfnHay6iB3OH-1jbFKxmuZ22Ayczl4DAQrBVeN1R3UJfqjhw-Mudw/s1600/picsym17.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="42" data-original-width="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvjoBBtDkJpjYhgP6_zGKg3d3PfYn1w1Milsg-U2AEgoOQv1wvs7SAgraVcPOfy4ShU0LJlLt1kci7JJwWradY5lOfnHay6iB3OH-1jbFKxmuZ22Ayczl4DAQrBVeN1R3UJfqjhw-Mudw/s1600/picsym17.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnncn8hpTuMV2sLOQK1h0wA7CiQvcCVy_v_rKI46e-diqg3fS5OcziyzWxk7EGKqu6AsScShIU78D4LmHEyg6Q52nUwBliO0nml44xaNqUSagPYVLOSEooqnJhmqTNPCrQQVhwibkPvEQ/s1600/picsym18.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="28" data-original-width="67" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnncn8hpTuMV2sLOQK1h0wA7CiQvcCVy_v_rKI46e-diqg3fS5OcziyzWxk7EGKqu6AsScShIU78D4LmHEyg6Q52nUwBliO0nml44xaNqUSagPYVLOSEooqnJhmqTNPCrQQVhwibkPvEQ/s1600/picsym18.png" /></a></div>
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<span style="white-space: pre;"> 3. </span>If rows were swapped, multiply the product of the diagonal by (-1) raised to the number of times rows were swapped.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGjljRE0bOIFJ2LFcRwaLpZSQVg-JvEIqPpDeBudnClStQDUXAPoONRDR-5ThMqNPrz3doTIXKxV7nmgxzv7pBky8FSATPbSW8spfqmh73X-4io79tb3HQ2tVAPbfZhB708eilSoEnzA0/s1600/picsym19.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="42" data-original-width="145" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGjljRE0bOIFJ2LFcRwaLpZSQVg-JvEIqPpDeBudnClStQDUXAPoONRDR-5ThMqNPrz3doTIXKxV7nmgxzv7pBky8FSATPbSW8spfqmh73X-4io79tb3HQ2tVAPbfZhB708eilSoEnzA0/s1600/picsym19.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsF0v10PtdHmZ1uMBwL8RlWJguN7Zviw0HA9dINfru_ftUXRG1YkEI2rTYajOyoAFpHNu2a6N2cvMQy3a7F5kZ5PT969jeTf0woThc7NZlzYaaI5PV0gwRoY-M3HHrofYGwiakDL0mQx8/s1600/picsym20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="28" data-original-width="56" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsF0v10PtdHmZ1uMBwL8RlWJguN7Zviw0HA9dINfru_ftUXRG1YkEI2rTYajOyoAFpHNu2a6N2cvMQy3a7F5kZ5PT969jeTf0woThc7NZlzYaaI5PV0gwRoY-M3HHrofYGwiakDL0mQx8/s1600/picsym20.png" /></a></div>
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As you can see, this is a lot of material to learn and remember. Don’t let it intimidate you! Once you start practicing, it will get easier. For more help or practice on this topic, check out Symbolab’s <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice/matrices-practice?subTopic=Determinant"><span style="color: #1155cc;">Practice</span></a></span></u>. Next blog post, I will talk about a shortcut for calculating the inverse of a 2x2 matrix, using its determinant.<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-29462194333705248812018-12-04T04:18:00.001-08:002022-06-08T07:56:48.735-07:00Advanced Math Solutions - Matrix Inverse Calculator, MatricesIn this blog post, I will talk about how to get the inverse of a matrix. Please refresh yourself on <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="http://blog.symbolab.com/2018/10/advanced-math-solutions-matrix-gauss.html"><span style="color: #1155cc;">RREF</span></a></span></u> before continuing, if you haven’t mastered this topic yet.<br />
<br />
<b>What is the inverse of a matrix?</b><br />
The inverse of a matrix is like the reciprocal of a number. The inverse of an <span class="mathquill-embedded-latex font16">n\times n</span> matrix <span class="mathquill-embedded-latex font16">A</span> is <span class="mathquill-embedded-latex font16">A^{-1}</span>,an <span class="mathquill-embedded-latex font16">n\times n</span> matrix, such that <span class="mathquill-embedded-latex font16">A⋅A^{-1}=I</span>, where <span class="mathquill-embedded-latex font16">I</span> is the identity matrix. Just like how the product of a number and its reciprocal equals 1, (<span class="mathquill-embedded-latex font16">\frac{1}{n}⋅n=1</span>), the product of a matrix and its inverse equals the identity matrix.<br />
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An identity matrix is an <span class="mathquill-embedded-latex font16">n\times n</span> matrix, where the main diagonal of the matrix is all 1s and everywhere else in the matrix is 0s.<br />
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Note: You can only calculate the inverse of a square (<span class="mathquill-embedded-latex font16">n\times n</span>) matrix.<br />
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There are a couple of methods used to find the inverse of a matrix. In this blog post, we will go over the method that Symbolab uses, which is one of the most common methods.<br />
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<b>How do you calculate the inverse of a matrix?</b><br />
<span style="white-space: pre;"> 1. </span>Augment the <span class="mathquill-embedded-latex font16">n\times n</span> matrix <span class="mathquill-embedded-latex font16">A</span> with the <span class="mathquill-embedded-latex font16">n\times n</span> identity matrix to create matrix <span class="mathquill-embedded-latex font16">(A|I)</span><br />
<blockquote class="tr_bq">
<ul>
<li>Augmenting a matrix means to create a matrix by appending the columns of two matrices</li>
<li>Augmenting the matrix allows you to perform the same elementary row operations on both sides of the matrix</li>
</ul>
</blockquote>
<span style="white-space: pre;"> 2. </span>Put the matrix on the left hand side of the augmented matrix in RREF<br />
<blockquote class="tr_bq">
<ul>
<li>The matrix on the left hand side will be converted to the identity matrix<span style="font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"> </span> </li>
</ul>
</blockquote>
<blockquote class="tr_bq">
<ul>
<li> Whatever
elementary row operations you do to the left matrix will be done to the matrix
on the right</li>
</ul>
</blockquote>
<span style="white-space: pre;"> 3. </span>The inverse matrix, <span class="mathquill-embedded-latex font16">A^{-1}</span>, is to the right of the augmented matrix<br />
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Doesn’t sound too complicated right? As long as you’ve master putting matrices in RREF, this should be a piece of cake. Let’s see some examples!<br />
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First example<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZyCMzJKlMnsYDah1enOyDPyyqHX-NzwuvjPE7eapHfntWH2L42Y7AGZblBh7wmPiNNx_NDpZmWW_z20cofwHzRqYhR2XlqZq_O3JFjJ7jAmqqO1-NUYjiPlyDbgj31exn_WC09yl7jQU/s1600/picsym2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="157" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZyCMzJKlMnsYDah1enOyDPyyqHX-NzwuvjPE7eapHfntWH2L42Y7AGZblBh7wmPiNNx_NDpZmWW_z20cofwHzRqYhR2XlqZq_O3JFjJ7jAmqqO1-NUYjiPlyDbgj31exn_WC09yl7jQU/s1600/picsym2.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Augment the matrix with the identity matrix<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgREQgEkAqNAo2NsBI0sJz3N-mF2MsOBVRILOawgBnFSToPZmy_OGsvyQ6MP3-6_50_jDo9AqtMuxD1UwoF3AWGPawPAK8BU9oN65F4u-TX1xEJKF7tZX1G1QLydmjcSnyfj4DZIIV742w/s1600/picsym3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="51" data-original-width="134" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgREQgEkAqNAo2NsBI0sJz3N-mF2MsOBVRILOawgBnFSToPZmy_OGsvyQ6MP3-6_50_jDo9AqtMuxD1UwoF3AWGPawPAK8BU9oN65F4u-TX1xEJKF7tZX1G1QLydmjcSnyfj4DZIIV742w/s1600/picsym3.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Put the matrix on the left in RREF<br />
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<span class="mathquill-embedded-latex font16">R_2-R_1→R_2</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKQqltjV1fhxn9SUXzasxsQenNeRo_CgCYmNZzmtRq9Dy2Vvt-4fWW8Ii_2xyudpSvZz5n9sDgYMGkiXb-V2Acg9nBwGD2YwandVK7rGnDCPDuu70feMQaQsUSH_9wodLrZ8JxyujvERE/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="181" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKQqltjV1fhxn9SUXzasxsQenNeRo_CgCYmNZzmtRq9Dy2Vvt-4fWW8Ii_2xyudpSvZz5n9sDgYMGkiXb-V2Acg9nBwGD2YwandVK7rGnDCPDuu70feMQaQsUSH_9wodLrZ8JxyujvERE/s1600/picsym4.png" /></a></div>
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<span class="mathquill-embedded-latex font16">\frac{1}{3} R_2→R_2</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1JCpyHPEmc1MvUsr1eam8Ds30HhWcp9whFEI3CwjhIecnmNrdVTmk9qA6UDkRYlvD7p8Nah5arjvmW404A0b4l56lAFRz5qt-aoCZgmBLcZB8SUrFrxVoY5I2xjOPPUoK8lljC0hix44/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="77" data-original-width="206" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1JCpyHPEmc1MvUsr1eam8Ds30HhWcp9whFEI3CwjhIecnmNrdVTmk9qA6UDkRYlvD7p8Nah5arjvmW404A0b4l56lAFRz5qt-aoCZgmBLcZB8SUrFrxVoY5I2xjOPPUoK8lljC0hix44/s1600/picsym5.png" /></a></div>
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<span class="mathquill-embedded-latex font16">R_1-R_2→R_1</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEFxenxBQRBN-bcH3jh98MhbA1ZAl3-wY8A0lL_bJbegMVScUPBah8KNxO7jEBzYcDpRPiQ-bMbF6GLgVG9-AoAtbjzkkb4J2-UtJvDJQW_zbdJbomJBuXfUx9SYhPDGbeYcnhjkzv5B4/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="96" data-original-width="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEFxenxBQRBN-bcH3jh98MhbA1ZAl3-wY8A0lL_bJbegMVScUPBah8KNxO7jEBzYcDpRPiQ-bMbF6GLgVG9-AoAtbjzkkb4J2-UtJvDJQW_zbdJbomJBuXfUx9SYhPDGbeYcnhjkzv5B4/s1600/picsym6.png" /></a></div>
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<span class="mathquill-embedded-latex font16">\frac{1}{2} R_1→R_1</span><br />
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<span style="white-space: pre;"> 3. </span>The inverse matrix is on the right of the augmented matrix<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrLN7pa8gPF9X837tL5j3XcXCPQFUqpJzlU37gSxs_eF2thU8NaxcmJc0FPaxHphHYMHNtiYbQXiTs5j-SzOav7VSwLrvjxy6Dwb3_6dbvOxyc1CwooZAmB6GOnq58doV4FFEKV8yipuQ/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="92" data-original-width="296" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrLN7pa8gPF9X837tL5j3XcXCPQFUqpJzlU37gSxs_eF2thU8NaxcmJc0FPaxHphHYMHNtiYbQXiTs5j-SzOav7VSwLrvjxy6Dwb3_6dbvOxyc1CwooZAmB6GOnq58doV4FFEKV8yipuQ/s1600/picsym8.png" /></a></div>
<blockquote class="tr_bq">
You can verify that this is the inverse of the matrix by multiplying the inverse of the matrix and the matrix together (see <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D2%261%5C%5C%202%264%5Cend%7Bpmatrix%7D%5Ccdot%5Cbegin%7Bpmatrix%7D%5Cfrac%7B2%7D%7B3%7D%26-%5Cfrac%7B1%7D%7B6%7D%5C%5C%20-%5Cfrac%7B1%7D%7B3%7D%26%5Cfrac%7B1%7D%7B3%7D%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">here</span></a></span></u>). If the product equals the identity matrix, then it is the inverse.</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUO4QgIpMg8HjGmAVibECjYivFAfjVy-qi6XE4A6FIMObHjCH2UWxs2RCfphRE1MiqqV3OCPG-KgGSlZce9cinTBxDyZOj18PAfy9Ie30fXuOmfJdz_rIICWIcXQ110DHDOaMX0Au1FrU/s1600/picsym9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="82" data-original-width="218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUO4QgIpMg8HjGmAVibECjYivFAfjVy-qi6XE4A6FIMObHjCH2UWxs2RCfphRE1MiqqV3OCPG-KgGSlZce9cinTBxDyZOj18PAfy9Ie30fXuOmfJdz_rIICWIcXQ110DHDOaMX0Au1FrU/s1600/picsym9.png" /></a></div>
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Next example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/inverse%20%5Cbegin%7Bpmatrix%7D4%264%262%5C%5C%20%200%262%264%5C%5C%208%262%262%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvsgMUB1wvt7jaXd8OR7LQ29ljla3CXg-nVaJUstRA0-d8BLJBwOxsMl3_joipSL-dMrxHlhYPXibMhlyRoUBFuHgLZK6yvSHegq5DE38eRvvByoLqdJQ-57hyphenhyphene3fQt29rMlH67ByQgxQ/s1600/picsym10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="79" data-original-width="189" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvsgMUB1wvt7jaXd8OR7LQ29ljla3CXg-nVaJUstRA0-d8BLJBwOxsMl3_joipSL-dMrxHlhYPXibMhlyRoUBFuHgLZK6yvSHegq5DE38eRvvByoLqdJQ-57hyphenhyphene3fQt29rMlH67ByQgxQ/s1600/picsym10.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Augment the matrix with the identity matrix<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz3LMetJ0Porec90jmRjH9ESQHYqwwbARjVlsLXpooImlpx6c9Sz5CNPcVmPYy92ndFSuaUJARClijU420I_j6tZ_XqfCI0R2ndyJVv85jDHCGC_umV604VhFOlxqBImm9VSriYq0dOXo/s1600/picsym11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="73" data-original-width="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz3LMetJ0Porec90jmRjH9ESQHYqwwbARjVlsLXpooImlpx6c9Sz5CNPcVmPYy92ndFSuaUJARClijU420I_j6tZ_XqfCI0R2ndyJVv85jDHCGC_umV604VhFOlxqBImm9VSriYq0dOXo/s1600/picsym11.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Put the matrix on the left in RREF<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUMG-A6WhFUYYZJu9VxAdabGikL3Kw6opSoTskqdTyn_-lPQFx0Ca9nXn6IIJxo6nRCAMs8yZnTndA7u01In8Ccv0pRtahJXaor9wB4NdmqWj6v0Y4yjt_0CKz3qsj6PY3OTwXW7BCnJ4/s1600/picsym12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="114" data-original-width="260" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUMG-A6WhFUYYZJu9VxAdabGikL3Kw6opSoTskqdTyn_-lPQFx0Ca9nXn6IIJxo6nRCAMs8yZnTndA7u01In8Ccv0pRtahJXaor9wB4NdmqWj6v0Y4yjt_0CKz3qsj6PY3OTwXW7BCnJ4/s1600/picsym12.png" /></a></div>
<blockquote class="tr_bq">
Since we’ve already gone over how to put a matrix in RREF in a previous blog post and in the first example, we won’t go over how to do this.</blockquote>
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<span style="white-space: pre;"> 3. </span>The inverse matrix is on the right of the augmented matrix<br />
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Last example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D1%262%260%262%5C%5C%202%260%26-1%260%5C%5C%20%200%26-1%26-2%261%5C%5C%20%202%262%260%261%5Cend%7Bpmatrix%7D%5E%7B-1%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgISrkDI_Hs3LxRpsBEDAnwdXTE6o06ijK_-xl1cIdbmU91WvUjeBM55BXfJeGI1-rshgTFvp5YQiJIZfdLW4j8SGyuoUjp59rgM-tFZH72RELRpc8KWGLx5cp6mutNrU_tW1XAZIvdM2U/s1600/picsym14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="110" data-original-width="199" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgISrkDI_Hs3LxRpsBEDAnwdXTE6o06ijK_-xl1cIdbmU91WvUjeBM55BXfJeGI1-rshgTFvp5YQiJIZfdLW4j8SGyuoUjp59rgM-tFZH72RELRpc8KWGLx5cp6mutNrU_tW1XAZIvdM2U/s1600/picsym14.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Augment the matrix with the identity matrix<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCOrM8nsOqNOESKaidpjAh5VJu8CpCbUBfLU0Lvwdd69SfhEx-jP0bhluYK_x2LOTSXtCeIhsq2n9pbv4bB52QGe45i41tIRnNVGlDczef9SNbDaFbNa2HwuVZZokR3TE_ON1kEVehkeE/s1600/picsym15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="85" data-original-width="261" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCOrM8nsOqNOESKaidpjAh5VJu8CpCbUBfLU0Lvwdd69SfhEx-jP0bhluYK_x2LOTSXtCeIhsq2n9pbv4bB52QGe45i41tIRnNVGlDczef9SNbDaFbNa2HwuVZZokR3TE_ON1kEVehkeE/s1600/picsym15.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Put the matrix on the left in RREF<br />
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<span style="white-space: pre;"> 3. </span>The inverse matrix is on the right of the augmented matrix<br />
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As you can see, as long as you know how to put a matrix in RREF, finding the inverse of a matrix doesn’t require too much additional work.<br />
<br />
It is important to make sure that you double check your answer by verifying the product of the matrix and its inverse is the identity matrix because it is very easy to make mistake while adding and subtracting rows.<br />
<br />
For more help or practice on this topic, check out Symbolab’s <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice"><span style="color: #1155cc;">Practice</span></a></span></u>.<br />
Next blog post, I’ll talk about what the determinant of a matrix is and how to calculate it.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-78243027320546058852018-10-23T03:36:00.002-07:002022-06-08T07:56:49.930-07:00Advanced Math Solutions - Matrix Gauss Jordan Reduction Calculator, Gauss Jordan EliminationIn our previous blog posts, we talked about Row Echelon Form (Gaussian Elimination). If you haven’t, please look <a href="http://blog.symbolab.com/2018/10/advanced-math-solutions-matrix-row.html" target="_blank">over it </a>before continuing with this blog post.<br />
<br />
In this blog post, we’ll talk about another advanced matrix topic that uses the same concepts, Gauss Jordan Elimination.<br />
<br />
The Gauss Jordan Elimination is a method of putting a matrix in row reduced echelon form (RREF), using elementary row operations, in order to solve systems of equations, calculate rank, calculate the inverse of matrix, and calculate the determinant of a matrix (we will cover this in the next few blog posts).<br />
<br />
RREF is when a matrix qualifies for the following four characteristics:<br />
<ul>
<li>Each non-zero row has 1, called a leading 1, as their first non-zero entry</li>
<li>Each column with a leading 1 has zeros in every other entry</li>
<li>As you move down the rows, the leading 1 moves to the right </li>
<li>All zero rows are at the bottom</li>
</ul>
<br />
Another thing to note: Unlike matrices in REF, matrices in RREF are unique.<br />
<br />
Here are examples of matrices in RREF:<br />
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Here are examples of matrices that aren’t in RREF:<br />
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You’ll use the same elementary row operations that you use to put a matrix in REF to put a matrix in RREF.<br />
<br />
Here are guidelines on how to put a matrix in RREF:<br />
<ol>
<li>Put the matrix in REF</li>
<li>If there are nonzero entries in the column of the leading coefficient in the first row, make them 0 by using the elementary row operations</li>
<li>If the leading coefficient in the first row is not a 1, make it a 1 by multiply the row by the reciprocal (this turns the leading coefficient into the leading 1)</li>
<li>Repeat steps 2-3 and replace “first” with “second”, then “third”, and so on</li>
</ol>
<br />
Gauss Jordan Elimination can be tricky the first few times, so I will walk you through 3 examples.<br />
<br />
First example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/gauss%20jordan%20%5Cbegin%7Bpmatrix%7D5%2610%5C%5C%20%201%264%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<span style="white-space: pre;"> </span>1. Put the matrix in REF<br />
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<span style="white-space: pre;"> </span>2. Get rid of the nonzero entry in the column of the leading coefficient in Row 2: <span class="mathquill-embedded-latex font16">R_1-5R_2→R_1</span><br />
<span class="mathquill-embedded-latex font16"><br /></span>
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<span style="white-space: pre;"> </span>3. Turn the leading coefficient into a 1 in Row 1: <span class="mathquill-embedded-latex font16">\frac{1}{5} R_1→R_1</span><br />
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<span style="white-space: pre;"> </span>4. Turn the leading coefficient into a 1 in Row 2: <span class="mathquill-embedded-latex font16">\frac{1}{2} R_2→R_2</span><br />
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Example 2 (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/gauss%20jordan%20%5Cbegin%7Bpmatrix%7D3%260%26-8%5C%5C%201%268%264%5C%5C%200%262%26%5Cfrac%7B5%7D%7B3%7D%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<span style="white-space: pre;"> 1. </span>Put the matrix in REF<br />
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<span style="white-space: pre;"> 2. </span>Turn the leading coefficient in Row 1 into 1: <span class="mathquill-embedded-latex font16">\frac{1}{3} R_1→R_1</span><br />
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<span style="white-space: pre;"> 3. </span>Turn the leading coefficient in Row 2 into 1: <span class="mathquill-embedded-latex font16">\frac{1}{8} R_2→R_2</span><br />
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In this problem there was no leading coefficient in Row 3, so we didn’t have to get rid of the entries in column 3.<br />
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Example 3 (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/gauss%20jordan%20%5Cbegin%7Bpmatrix%7D2%261%260%264%5C%5C%200%260%262%265%5C%5C%203%261%260%266%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<span style="white-space: pre;"> 1. </span>Put the matrix in REF<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSTzx7iqnm9Iq6kshTA6MgATn4HUyvsgSYDylaeMcGNmZdiB37q8lKHkccD_6QaTjyu3jwrkfQtUedAEqF-nKMlZoaRokAVin6XwprpfA17WXYA7jAiBdR89c7_dxgeVZ-s3tBEOsAl2U/s1600/picsym16.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="105" data-original-width="169" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSTzx7iqnm9Iq6kshTA6MgATn4HUyvsgSYDylaeMcGNmZdiB37q8lKHkccD_6QaTjyu3jwrkfQtUedAEqF-nKMlZoaRokAVin6XwprpfA17WXYA7jAiBdR89c7_dxgeVZ-s3tBEOsAl2U/s1600/picsym16.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Get rid of the nonzero entry in the column of the leading coefficient in Row 2: <span class="mathquill-embedded-latex font16">R_1-3R_2→R_1</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEie_dD1VddfgGxDBqisZgU_vj3OIstJix1LRTrE4IjoeHwqpit9yHarfpIQ1NJFvdTD_BMlLPwRbYiCTVgcYZY3IS8TUsKETARgFr4bhIMYfDH_FUizziACF2Q4T_TPIou-KcN4lNP4QT4/s1600/picsym17.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="103" data-original-width="167" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEie_dD1VddfgGxDBqisZgU_vj3OIstJix1LRTrE4IjoeHwqpit9yHarfpIQ1NJFvdTD_BMlLPwRbYiCTVgcYZY3IS8TUsKETARgFr4bhIMYfDH_FUizziACF2Q4T_TPIou-KcN4lNP4QT4/s1600/picsym17.png" /></a></div>
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<span style="white-space: pre;"> 3. </span>Turn the leading coefficient in Row 1 into 1: <span class="mathquill-embedded-latex font16">\frac{1}{3} R_1→R_1</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnwhlO1BBV6OUWi6ak9Ce2f62y84X1esr4W56p0a7vMW40f9tVZHG0gz2VQ1nRyxSvCkFwZpYJ9uf9vPxcjQEJukMIWGNBdjtluenprtV9B9BLJPk3tWsW77KGQviJW_crv-34b2lv0Qk/s1600/picsym18.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="101" data-original-width="167" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnwhlO1BBV6OUWi6ak9Ce2f62y84X1esr4W56p0a7vMW40f9tVZHG0gz2VQ1nRyxSvCkFwZpYJ9uf9vPxcjQEJukMIWGNBdjtluenprtV9B9BLJPk3tWsW77KGQviJW_crv-34b2lv0Qk/s1600/picsym18.png" /></a></div>
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<span style="white-space: pre;"> 4. </span>Turn the leading coefficient in Row 2 into 1: <span class="mathquill-embedded-latex font16">3R_2→R_2</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi73sVHrBOHrGCBFRZTCLD9a0d8qMYJo3sBqvnAbm2oqQFJhb3XU0z1ZUh0kDvI5jZVDTamtdv_Z-v00lwZgGaSMIA6e0DTejhHnn8VtWcr00VP4IexUX9EH7cXcgf9IQuCe14KIn88O_0/s1600/picsym19.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="78" data-original-width="165" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi73sVHrBOHrGCBFRZTCLD9a0d8qMYJo3sBqvnAbm2oqQFJhb3XU0z1ZUh0kDvI5jZVDTamtdv_Z-v00lwZgGaSMIA6e0DTejhHnn8VtWcr00VP4IexUX9EH7cXcgf9IQuCe14KIn88O_0/s1600/picsym19.png" /></a></div>
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<span style="white-space: pre;"> 5. </span>Turn the leading coefficient in Row 3 into 1: <span class="mathquill-embedded-latex font16">\frac{1}{2} R_3→R_3</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLpov0SmgeECz1oPwSQaz-vmgmmiEfj8Ho0pH9LMMZTgepTSYQv_OHQhXk6-ktYYBATXL8O_FOUQrL0e7FQxWFQ062yLCKqPsxW12Ou-9DHuCqSCcDLs5XoZfdWrmXiz06MR_4wTL8Ea4/s1600/picsym20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="103" data-original-width="167" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLpov0SmgeECz1oPwSQaz-vmgmmiEfj8Ho0pH9LMMZTgepTSYQv_OHQhXk6-ktYYBATXL8O_FOUQrL0e7FQxWFQ062yLCKqPsxW12Ou-9DHuCqSCcDLs5XoZfdWrmXiz06MR_4wTL8Ea4/s1600/picsym20.png" /></a></div>
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Putting a matrix in RREF can require a lot of steps, so it is important to document what you are doing in each row. This will help you look over your work and remember what you did. Just like putting a matrix in REF, practice will help you get better at putting a matrix in RREF.<br />
<br />
For more help or practice on this topic, visit Symbolab’s <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice"><span style="color: #1155cc;">Practice</span></a></span></u>.<br />
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Until next time,<br />
<br />
Leah.<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-75505372967669930442018-10-09T04:23:00.002-07:002022-06-08T07:56:51.567-07:00Advanced Math Solutions - Matrix Row Echelon Calculator, Gaussian Elimination (Row Echelon Form)In our previous blog posts, we talked about the Matrix basics. Now, we are ready to talk about a more advanced matrix topic, Gaussian Elimination (also known as row echelon form).<br />
<br />
The Gaussian Elimination, is a method of putting a matrix in row echelon form (REF), using elementary row operations.<br />
<br />
REF is when a matrix qualifies for the following two characteristics:<br />
<ul>
<li>Each nonzero row has a leading coefficient (the first nonzero entry) that is to the right of the leading coefficient of the row above it</li>
</ul>
<ul><ul>
<li>There can’t be any nonzero entries below the leading coefficient in the leading coefficient’s column</li>
</ul>
</ul>
<ul>
<li>All zero rows are at the bottom</li>
</ul>
<br />
Note: A matrix in REF is not unique, so you may have a slightly different solution.<br />
<br />
Here are examples of REF:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs3xvL6Td6qOvd_6FIc0ytQZB4XMvmA7iENUWoDEAzCvTXzevXihgPW7SGvFMGtalYfM2LIuI7t5MS6aR3tdlLTkvYdohUSfK75RA8wlQmklMoZcKPy-fSaFMEPUvP8e3BjNo-i2I467c/s1600/pic1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="87" data-original-width="99" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs3xvL6Td6qOvd_6FIc0ytQZB4XMvmA7iENUWoDEAzCvTXzevXihgPW7SGvFMGtalYfM2LIuI7t5MS6aR3tdlLTkvYdohUSfK75RA8wlQmklMoZcKPy-fSaFMEPUvP8e3BjNo-i2I467c/s1600/pic1.png" /></a></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZdp6rS2JdxRygOlKHPQJm0VOoYlE1gP18ejPjTQvrY0-2WSlrgtfYd4vatS9ccNW5nWw2OHD6Jguzsg1XT0uB3P-zPw307V557uajcIW-cqiHv3o4FkB5H75LmWyZ-OE-fWEq0cey6g8/s1600/pic2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="80" data-original-width="129" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZdp6rS2JdxRygOlKHPQJm0VOoYlE1gP18ejPjTQvrY0-2WSlrgtfYd4vatS9ccNW5nWw2OHD6Jguzsg1XT0uB3P-zPw307V557uajcIW-cqiHv3o4FkB5H75LmWyZ-OE-fWEq0cey6g8/s1600/pic2.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6o4d04XpanITc2dEkPP9DAsi2_-jDMWm8Y8hVihB7tg-MaaaR6OhCvEZXmG5qv_-i6B7oDDyXAFF1x6mKt4BltT6haz0ZQXVDq-RLzdeE7jhpMZU0EqcY28FijyV-s0Ffahd6XtgACK0/s1600/pic3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="80" data-original-width="99" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6o4d04XpanITc2dEkPP9DAsi2_-jDMWm8Y8hVihB7tg-MaaaR6OhCvEZXmG5qv_-i6B7oDDyXAFF1x6mKt4BltT6haz0ZQXVDq-RLzdeE7jhpMZU0EqcY28FijyV-s0Ffahd6XtgACK0/s1600/pic3.png" /></a><br />
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Here are examples of matrices that aren’t in REF:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg32Dg2lAdqDmFjVSIb1YsM1Pwcp0nHG_u_Gtpnt4nJ5rBG09P5hbFG5_yw_rY7suaTpx_qdQBXWYVD4y6ZWcc6JYoYNo05u9CpU3a4ODStbBVbEVPsMMTGSwJjeOKOZfcxXCsCoqtuJRk/s1600/pic4.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="83" data-original-width="104" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg32Dg2lAdqDmFjVSIb1YsM1Pwcp0nHG_u_Gtpnt4nJ5rBG09P5hbFG5_yw_rY7suaTpx_qdQBXWYVD4y6ZWcc6JYoYNo05u9CpU3a4ODStbBVbEVPsMMTGSwJjeOKOZfcxXCsCoqtuJRk/s1600/pic4.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYZfofPS-wPmZiqUxBQKYCEaiGL8f94FrrZUtMVhikEjEr8yyaBKf1_9SuKCxaGtn4VQUmtiWfomrXFWKaUjtyFElDa3cR4xPTkZKlX0AKqupsd0WsQHF2lcK2Q0vbWeb53ex81SFauj8/s1600/pic5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="84" data-original-width="98" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYZfofPS-wPmZiqUxBQKYCEaiGL8f94FrrZUtMVhikEjEr8yyaBKf1_9SuKCxaGtn4VQUmtiWfomrXFWKaUjtyFElDa3cR4xPTkZKlX0AKqupsd0WsQHF2lcK2Q0vbWeb53ex81SFauj8/s1600/pic5.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguwqVgD_GTvd7LUya4TEWOK-d18PwWL49aqvw-nqn-VlkrYOXMTjw0ufdnKTiUfadd1sDYfaEgWomlNb4ssIS47k1Xd6Vw9EmD2BDi2CK1WBV_0yjsL7T3Wq9r7YORdHVoX09IdeNH9-E/s1600/pic6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="82" data-original-width="127" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguwqVgD_GTvd7LUya4TEWOK-d18PwWL49aqvw-nqn-VlkrYOXMTjw0ufdnKTiUfadd1sDYfaEgWomlNb4ssIS47k1Xd6Vw9EmD2BDi2CK1WBV_0yjsL7T3Wq9r7YORdHVoX09IdeNH9-E/s1600/pic6.png" /></a></div>
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The elementary row operations you’ll use to put your matrix in REF are:<br />
<ul>
<li>Switch any two rows</li>
<li>Multiply each entry in a row by a non-zero constant</li>
<li>Replace a row by the sum/difference of the row itself and another row, where it’s entries are multiplied by a non-zero constant</li>
</ul>
<br />
Here is a guideline on how to put a matrix in REF:<br />
<ol>
<li>Move all zero rows to the bottom</li>
<li>Begin at the first row</li>
<li>If the first entry is a zero, switch the row with a row below it that has non-zero entry in the first column</li>
<li>If there are nonzero entries below the leading coefficient of the first row in the same column, cancel the entries by subtracting multiples of the the first row to the other rows (this will result in a zero entry)</li>
<li>Repeat steps 2 - 4 and replace “first” with “second”, then “third”, and so on until you can’t do anything more</li>
</ol>
<br />
This topic can be hard to understand at first, so let’s see some examples to better understand.<br />
<br />
First example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/row%20echelon%20%5Cbegin%7Bpmatrix%7D6%2612%5C%5C%20%20%20%20%202%269%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRbY7o1Fe9MG0RKOzLkhQgUR2WB_wCyF14uT65fbwDYKL6ATil4vEYMbf_NI9EFMj3INTpsnIAQBrPUVVl0OlJGJmD8DZD8NWofU8sC0sIfsE_gg7_o4WEvmgXrabU0Z8i5FgmVokE36U/s1600/pic7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="58" data-original-width="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRbY7o1Fe9MG0RKOzLkhQgUR2WB_wCyF14uT65fbwDYKL6ATil4vEYMbf_NI9EFMj3INTpsnIAQBrPUVVl0OlJGJmD8DZD8NWofU8sC0sIfsE_gg7_o4WEvmgXrabU0Z8i5FgmVokE36U/s1600/pic7.png" /></a></div>
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<span style="white-space: pre;"> </span>1. Cancel the leading coefficient in Row 2: <span class="mathquill-embedded-latex font16">R_2-\frac{1}{3} R_1→ R_2</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2voaAPfOgEYaS8RfsZcX9f3egXO7utiZ6hxAYCEHoRsJ7TQVyuqgqfZJB_nykpTwWMpM55A_jE3tmAVeWNX2EN7R_OHPkKssBQUEEemeulTiNkWb4vFRHueVtIfxvlLeSbtj2suigzRE/s1600/pic8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="102" data-original-width="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2voaAPfOgEYaS8RfsZcX9f3egXO7utiZ6hxAYCEHoRsJ7TQVyuqgqfZJB_nykpTwWMpM55A_jE3tmAVeWNX2EN7R_OHPkKssBQUEEemeulTiNkWb4vFRHueVtIfxvlLeSbtj2suigzRE/s1600/pic8.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Simplify Row 2<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs1oq5g7-zVo4lHKzXegysLzxbbY22vBVcQAaO5-BWUHEhZnfrVIJDxXONKUNce3xmilYZxERz7iGpPP3QOLw-1IztGx3JSrKlz6FH8TsXVkp2sjbT57eBOH3jQJAmRvjHjvGXTFmzUa0/s1600/pic9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="72" data-original-width="147" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs1oq5g7-zVo4lHKzXegysLzxbbY22vBVcQAaO5-BWUHEhZnfrVIJDxXONKUNce3xmilYZxERz7iGpPP3QOLw-1IztGx3JSrKlz6FH8TsXVkp2sjbT57eBOH3jQJAmRvjHjvGXTFmzUa0/s1600/pic9.png" /></a></div>
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We got rid of (canceled out) the 2 in Row 2 because there can’t be any entries below the leading coefficient in Row 1. We were able to do this using the elementary row operations. Multiplying Row 1 by ⅓, in order to turn 6 into 2, and then subtracting Row 2 by ⅓ Row 1.<br />
<br />
Next example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/row%20echelon%20%5Cbegin%7Bpmatrix%7D2%268%260%5C%5C%201%260%266%5C%5C%200%263%262%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDkPRPKqTbMjSfEIokuhy6M90r_ZYSXN3tHWPn3uR4v1NtcoKFRjKsf_3tGJito_6YAqlLMMbeD3p8Fp0S-UZcZAonL09L5XYxb2gOkjQL4u5K797kv4VUdvfO0fjkbYHolhwx0Q4U_mA/s1600/pic10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="81" data-original-width="218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDkPRPKqTbMjSfEIokuhy6M90r_ZYSXN3tHWPn3uR4v1NtcoKFRjKsf_3tGJito_6YAqlLMMbeD3p8Fp0S-UZcZAonL09L5XYxb2gOkjQL4u5K797kv4VUdvfO0fjkbYHolhwx0Q4U_mA/s1600/pic10.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Cancel the leading coefficient in Row 2: <span class="mathquill-embedded-latex font16">R_2-\frac{1}{2} R_1 →R_2</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsMNoc-jYLJTNHeZzK6kRJm3-XAzr5I0oVlF0oKy1ZPIMVTHrwuW0g6PVxEoWrT1psaqNOXH06xsJZHSS-DIiE_nHMIFhTGwbdAhZhpQoIiqmjlNTlDRW7VmI9awriw0tF_ZkMD1-O9TQ/s1600/pic11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="119" data-original-width="264" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsMNoc-jYLJTNHeZzK6kRJm3-XAzr5I0oVlF0oKy1ZPIMVTHrwuW0g6PVxEoWrT1psaqNOXH06xsJZHSS-DIiE_nHMIFhTGwbdAhZhpQoIiqmjlNTlDRW7VmI9awriw0tF_ZkMD1-O9TQ/s1600/pic11.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizd9_37aG3-mM7Cw9ZTMUSu8tiCBf1w4MJl-CEBVEp272_UYOxYo_HfqDeU5nHxM7-uI2Y3-vCUQboURIWxNzFeEPT69q-YOa0tLYk-S1JYND3OfhGIuzWUMuw11xFVSzGuxC2mgdtgHM/s1600/pic12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="88" data-original-width="154" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizd9_37aG3-mM7Cw9ZTMUSu8tiCBf1w4MJl-CEBVEp272_UYOxYo_HfqDeU5nHxM7-uI2Y3-vCUQboURIWxNzFeEPT69q-YOa0tLYk-S1JYND3OfhGIuzWUMuw11xFVSzGuxC2mgdtgHM/s1600/pic12.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Cancel the leading coefficient in Row 3: <span class="mathquill-embedded-latex font16">R_3-(-\frac{3}{4} R_2) →R_3</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5qqnMdR1rGGFe9l6cwYUjHqwoBgSD6BoAnY6-SnDxiBYT8V8mUhqqbkRX5t2KG7m3Ae-UTCPIwBxjU9n_OoLP3XN0FE32lkZ9ZhjX2TZoBav1Nf-JTV-9QN8mVF6mFjh43lb_z9X8zuo/s1600/pic13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="102" data-original-width="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5qqnMdR1rGGFe9l6cwYUjHqwoBgSD6BoAnY6-SnDxiBYT8V8mUhqqbkRX5t2KG7m3Ae-UTCPIwBxjU9n_OoLP3XN0FE32lkZ9ZhjX2TZoBav1Nf-JTV-9QN8mVF6mFjh43lb_z9X8zuo/s1600/pic13.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNFszF3cZTFlUFCvsyBOnlLJC9sdwUh8rAyNhOXYeAKWUq7u-QD0W-ByYjKRWNTd3qGd6-GS36j6puZRrIQlcmlezxKVBH470lBVWDuWzH6W8I3D_3IEB1Szp-m7He1_ncOIwbU-tzCcM/s1600/pic14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="103" data-original-width="165" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNFszF3cZTFlUFCvsyBOnlLJC9sdwUh8rAyNhOXYeAKWUq7u-QD0W-ByYjKRWNTd3qGd6-GS36j6puZRrIQlcmlezxKVBH470lBVWDuWzH6W8I3D_3IEB1Szp-m7He1_ncOIwbU-tzCcM/s1600/pic14.png" /></a></div>
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Last example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/row%20echelon%20%5Cbegin%7Bpmatrix%7D2%260%266%262%5C%5C%20%20%201%260%264%269%5C%5C%20%201%266%269%264%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1jUX6th_GzxDX8ntqnSt_vA_8whyphenhyphenN1VkQU2-LzZ_hGAn6wyMszGqwyqDlokmKKYJ8pkpWwd1XKP_LyG8fw-xNxC35ONv13YfisveTSo7Ia7vlWoNciRn4H5dzXojTW3RQVPkhyfI4P54/s1600/pic15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="82" data-original-width="261" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1jUX6th_GzxDX8ntqnSt_vA_8whyphenhyphenN1VkQU2-LzZ_hGAn6wyMszGqwyqDlokmKKYJ8pkpWwd1XKP_LyG8fw-xNxC35ONv13YfisveTSo7Ia7vlWoNciRn4H5dzXojTW3RQVPkhyfI4P54/s1600/pic15.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Cancel the leading coefficient in Row 2: <span class="mathquill-embedded-latex font16">R_2-\frac{1}{2} R_1 →R_2</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJZvPwnAdpnJJFq6FvorIKf0aVfz0ewvk6mLMCXoILZVUaCALgEms3DW_Pvsw6-YRD_9Xg6-_gP8KXIJRA9xLh1llWtZXfJDspMAIla4SGNm7tu8IiZFG-lMdGpgXLGXtXbL-Cn4yFjC8/s1600/pic16.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="99" data-original-width="327" height="96" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJZvPwnAdpnJJFq6FvorIKf0aVfz0ewvk6mLMCXoILZVUaCALgEms3DW_Pvsw6-YRD_9Xg6-_gP8KXIJRA9xLh1llWtZXfJDspMAIla4SGNm7tu8IiZFG-lMdGpgXLGXtXbL-Cn4yFjC8/s320/pic16.png" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQ4zLvOIFA5Ilgt_7qP_9LjY7ZVEGDZj-iPg5VY76dNAEOOKTOjqaxfJSLPD6l81jb1aUCwhUYhcjQn7v7evbSQzOTRCIrMGk_xtdW6poBtWSKWcAOzPj4W3HT-2LZ_J5M7PownQ0AnFk/s1600/pic17.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="89" data-original-width="181" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQ4zLvOIFA5Ilgt_7qP_9LjY7ZVEGDZj-iPg5VY76dNAEOOKTOjqaxfJSLPD6l81jb1aUCwhUYhcjQn7v7evbSQzOTRCIrMGk_xtdW6poBtWSKWcAOzPj4W3HT-2LZ_J5M7PownQ0AnFk/s1600/pic17.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Cancel the leading coefficient in Row 3: <span class="mathquill-embedded-latex font16">R_3-\frac{1}{2} R_1→R_3</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTL3kcrU0Sh5Y9OwSNAypz9AKh5Pu8_f1Xoz3WPFki-KymaDQLKGKajvIp4JhYOmlrQoOHFcwGKzW7v-z6bh4Aq4UepeOjVv5OiWFP154UvlUC32zwtzRQbx9Ks0PzK4KDm0Z0qvNeFRE/s1600/pic18.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="99" data-original-width="318" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTL3kcrU0Sh5Y9OwSNAypz9AKh5Pu8_f1Xoz3WPFki-KymaDQLKGKajvIp4JhYOmlrQoOHFcwGKzW7v-z6bh4Aq4UepeOjVv5OiWFP154UvlUC32zwtzRQbx9Ks0PzK4KDm0Z0qvNeFRE/s1600/pic18.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3-CKt41TczY8zY3OwH_Cd-wpejn5NQHmaIe64kwQV5fThWQbKEIwIjQtpo1OQ3pOTGcuYZV-oqQ0O0HtC6rM7j36UX6DTZFjfbJfxGCWpy1PPvFVdvluVqyd5kWjh7fLnJYNvXVV2EUs/s1600/pic19.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="86" data-original-width="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3-CKt41TczY8zY3OwH_Cd-wpejn5NQHmaIe64kwQV5fThWQbKEIwIjQtpo1OQ3pOTGcuYZV-oqQ0O0HtC6rM7j36UX6DTZFjfbJfxGCWpy1PPvFVdvluVqyd5kWjh7fLnJYNvXVV2EUs/s1600/pic19.png" /></a></div>
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<span style="white-space: pre;"> 3. </span>Since the second entry is a zero, switch the row with a row below it that has non-zero entry in the second column: <span class="mathquill-embedded-latex font16">R_3↔R_2</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgkDQrj08xmf4Dx3rrwy0GhMhmSE_N0ltTh87HWdi96rmjPsyWvR4W4uRi7qxo52BqbGMFDs82kcAR3zn51jOzP4eZUdcxn_fqYaZj1BjYJSXaeuaTP22TRxgGcjPoEVhYzqmjvOmhkRE/s1600/pic20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="79" data-original-width="165" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgkDQrj08xmf4Dx3rrwy0GhMhmSE_N0ltTh87HWdi96rmjPsyWvR4W4uRi7qxo52BqbGMFDs82kcAR3zn51jOzP4eZUdcxn_fqYaZj1BjYJSXaeuaTP22TRxgGcjPoEVhYzqmjvOmhkRE/s1600/pic20.png" /></a></div>
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Putting matrices in REF can be tricky at first, but once you’ve done a handful of practice problems, it will come to you much easier. Check out Symbolab’s <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/matrix-row-echelon-calculator"><span style="color: #1155cc;">Row Echelon Calculator</span></a></span></u> to help you better understand this topic.<br />
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Until next time,<br />
<br />
Leah<br />
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Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-9036220970946030312018-09-30T07:09:00.001-07:002022-06-08T07:56:52.779-07:00Advanced Math Solutions - Matrix Trace Calculator, Matrix TraceIn today’s blog post, we will go over how to calculate the trace of a matrix. However, first, it is important to go over what the diagonal of a matrix is.<br />
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The <b>main diagonal</b> of an <span class="mathquill-embedded-latex font16">n</span> x <span class="mathquill-embedded-latex font16">n</span> matrix consists of entries whose row number is the same as its column number. In an <span class="mathquill-embedded-latex font16">n</span> x <span class="mathquill-embedded-latex font16">n</span> matrix, the diagonal is <span class="mathquill-embedded-latex font16">a_(1,1),a_(2,2),...,a_(n,n)</span>.<br />
Below is a matrix with its diagonal circled.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs7ZCmSVS2jiEzp4fhkTgf9u5k69eiyUV0mTSK9mBYWUlGhJXYq6B-kUacmzo0Qat8-Vyr_P1AfHku7wuS-AhJFHsGM2R2d24GasbL3wAUsLrapT5dBqJv121tY4MnpXujDd0BItFLMIc/s1600/pic1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="91" data-original-width="107" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs7ZCmSVS2jiEzp4fhkTgf9u5k69eiyUV0mTSK9mBYWUlGhJXYq6B-kUacmzo0Qat8-Vyr_P1AfHku7wuS-AhJFHsGM2R2d24GasbL3wAUsLrapT5dBqJv121tY4MnpXujDd0BItFLMIc/s1600/pic1.png" /></a></div>
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The <b>trace</b> of an <span class="mathquill-embedded-latex font16">n</span> x <span class="mathquill-embedded-latex font16">n</span> matrix is the sum of all the entries on the main diagonal. The trace of a matrix, <span class="mathquill-embedded-latex font16">A</span>, is denoted <span class="mathquill-embedded-latex font16">\tr(A)</span>. (Note: In order to calculate the trace of a matrix, the matrix must have the same number of rows and columns. Otherwise, there is no main diagonal.)<br />
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Let’s see some examples to better understand.<br />
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First example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Ctr%5Cbegin%7Bpmatrix%7D4%268%267%5C%5C%2010%2613%265%5C%5C%207%2620%266%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidZQWIaXxIN2aabYwTegbwjRRCbbNq_O2AeYIx4xeJCG8ZDKCbxT-erOsV2NChcmJz71ldWGE4xORp3-GEgWFK9ENQKQY0fxF-EerNEOa7l7s4I0DsMzLh__a88_CeM4EwZTkZno_zAgs/s1600/pic2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="83" data-original-width="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidZQWIaXxIN2aabYwTegbwjRRCbbNq_O2AeYIx4xeJCG8ZDKCbxT-erOsV2NChcmJz71ldWGE4xORp3-GEgWFK9ENQKQY0fxF-EerNEOa7l7s4I0DsMzLh__a88_CeM4EwZTkZno_zAgs/s1600/pic2.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Identify the main diagonal<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8NWhRn27Xg3QjOlgpmkVnofzivHYzC2Ydk6ZKVePhx9kmaTpi9ai-e7WCHkLSiUT4YB3G4f2G8px4U5E3lnSqb93YBjLStr6s9XO3o9OJwwxF2h8cD5k3HPa22J-pE2gNiZWlCAwp19A/s1600/pic3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="86" data-original-width="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8NWhRn27Xg3QjOlgpmkVnofzivHYzC2Ydk6ZKVePhx9kmaTpi9ai-e7WCHkLSiUT4YB3G4f2G8px4U5E3lnSqb93YBjLStr6s9XO3o9OJwwxF2h8cD5k3HPa22J-pE2gNiZWlCAwp19A/s1600/pic3.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Sum all the entries on the main diagonal<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA2TSakSWkAulgAHoovOcrMBoNri4aPO757cHiGnBQEiv00H6rUdDd3iPQ5ALkYjVZ0vL-DFL8tKW_ObiynN6vY5wDqZK4ZZwW3E9k_vLpOIj91cUW_xUB_x92O3h8UutJcK2Yf3UNR9I/s1600/pic4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="34" data-original-width="117" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA2TSakSWkAulgAHoovOcrMBoNri4aPO757cHiGnBQEiv00H6rUdDd3iPQ5ALkYjVZ0vL-DFL8tKW_ObiynN6vY5wDqZK4ZZwW3E9k_vLpOIj91cUW_xUB_x92O3h8UutJcK2Yf3UNR9I/s1600/pic4.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCXbzRsnZe0IkMTVYTw7z9ZPK92ixVJxf7Ud3haRXuQaldnsfseVJLs_b5zFwFP1DCAo3Hp17mY3eOYTKYQGiiTMCFFD9FI-blN60P7JpO2I9zqIvZ1w6spnNGA9agxenPdDanu5AG5Ok/s1600/pic5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="23" data-original-width="53" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCXbzRsnZe0IkMTVYTw7z9ZPK92ixVJxf7Ud3haRXuQaldnsfseVJLs_b5zFwFP1DCAo3Hp17mY3eOYTKYQGiiTMCFFD9FI-blN60P7JpO2I9zqIvZ1w6spnNGA9agxenPdDanu5AG5Ok/s1600/pic5.png" /></a></div>
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Next example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Ctr%5Cbegin%7Bpmatrix%7D4%267%2610%2619%5C%5C%20%20%2020%26-1%265%267%5C%5C%20%20%206%261%2625%264%5C%5C%20%20%204%268%262%265%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRTjVcSmW7hh9vWiFOY1bT5INrOKPgnD0oJ4U6dmaH2yxuXI0AypN9Pv_d0phqAeu1-D5CMUq0xKiM1Vu8Z7dG76cE_66ugDZMhKw_wUAyCuu9mqYAVsnfAYNgOZc_oGplXDJCboJNiBs/s1600/pic6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="109" data-original-width="221" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRTjVcSmW7hh9vWiFOY1bT5INrOKPgnD0oJ4U6dmaH2yxuXI0AypN9Pv_d0phqAeu1-D5CMUq0xKiM1Vu8Z7dG76cE_66ugDZMhKw_wUAyCuu9mqYAVsnfAYNgOZc_oGplXDJCboJNiBs/s1600/pic6.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Identify the main diagonal<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimEVMCENFyAkgbMl4N5yMQhv4piCbZ1YjWKGB8WG8xRvTcgiMH4IHyrKQlItiCSPmgTN3Wj66-fqSKZPjsDCqwL1pbmPN9_cBub3eKGzWDftd7QwjRXi93stcCyZWYJB3ncH0xnv9iMcY/s1600/pic7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="103" data-original-width="187" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimEVMCENFyAkgbMl4N5yMQhv4piCbZ1YjWKGB8WG8xRvTcgiMH4IHyrKQlItiCSPmgTN3Wj66-fqSKZPjsDCqwL1pbmPN9_cBub3eKGzWDftd7QwjRXi93stcCyZWYJB3ncH0xnv9iMcY/s1600/pic7.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Sum all of the entries on the main diagonal<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjN7x_v9GyRlaVvDchf7jP5goVQ_7b_lDuFh9NKDVLjAMm-OCTRrxFUd08UsjTPptuX7uqb8JTO5OjIshiuIMbVxDNDFgPkH_C0ocOfojTMHyFTFTpbnnHA5yHOuiSqtVfEw0OVf-_Ex2A/s1600/pic8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="29" data-original-width="192" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjN7x_v9GyRlaVvDchf7jP5goVQ_7b_lDuFh9NKDVLjAMm-OCTRrxFUd08UsjTPptuX7uqb8JTO5OjIshiuIMbVxDNDFgPkH_C0ocOfojTMHyFTFTpbnnHA5yHOuiSqtVfEw0OVf-_Ex2A/s1600/pic8.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgupMTSFfULKOO7LdfhgmeX9mbBvatioDBQXqmq58b5Q5H_KhN0wkd6SaAf0q_pfJ6_w6UtHvfX8CCJY7cQq_NLzjYD4d9o7Q0dcWuuB96S9FkdDgn9_6EKqZDoVBo_LeKADTcHuAAyvPs/s1600/pic9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="23" data-original-width="52" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgupMTSFfULKOO7LdfhgmeX9mbBvatioDBQXqmq58b5Q5H_KhN0wkd6SaAf0q_pfJ6_w6UtHvfX8CCJY7cQq_NLzjYD4d9o7Q0dcWuuB96S9FkdDgn9_6EKqZDoVBo_LeKADTcHuAAyvPs/s1600/pic9.png" /></a></div>
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Last example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Ctr%5Cbegin%7Bpmatrix%7D24%260%261%5C%5C%208%26-13%2610%5C%5C%207%268%26-5%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<span style="white-space: pre;"> 1. </span>Identify the main diagonal<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKwx_rwc0S5CB6L97ff4uWKTAnntoMCVtkqzPjeLCXUf7V7C-RxZqoTyEpE-5KdK5qKj1qy51jMycamjgvTnjU5YINo1srseDjF-cDnOMWTnFWWnPOX58enLkOZZ1dWcXtuQ4kqLr1iFw/s1600/pic11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="79" data-original-width="151" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKwx_rwc0S5CB6L97ff4uWKTAnntoMCVtkqzPjeLCXUf7V7C-RxZqoTyEpE-5KdK5qKj1qy51jMycamjgvTnjU5YINo1srseDjF-cDnOMWTnFWWnPOX58enLkOZZ1dWcXtuQ4kqLr1iFw/s1600/pic11.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Sum all of the entries on the main diagonal<br />
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Calculating the trace of a matrix is pretty easy. Just make sure to remember what the definition of the trace of a matrix is. For more help or practice on this topic, check out Symbolab’s <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice"><span style="color: #1155cc;">Practice</span></a></span></u>, which has more practice problems and quizzes.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-26093734307351021672018-06-18T05:35:00.001-07:002022-06-08T07:56:53.493-07:00High School Math Solutions - Matrix Multiply Calculator, Matrix Scalar Multiplication<a href="http://blog.symbolab.com/2018/06/advanced-math-solutions-matrix-multiply.html" target="_blank">Last blog post</a>, we talked about how to add and subtract matrices. Now, we will start getting into multiplication for matrices. There are two types: scalar multiplication and matrix multiplication. In this blog post, we will talk about the simpler of the two, scalar multiplication.<br />
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<b>Scalar multiplication</b> is when you multiply a matrix by a value, called a <b>scalar</b>. In scalar multiplication, you multiply each element of the matrix by the scalar.<br />
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Here is what scalar multiplication looks like:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi97DsITJiu2o4Sd4yMlELRHl4o1eea9chWtHmHcs3zo3H03VNj9l4UQUUm2VE2fUhsXqPMPzEUx5Sy39ce7nC4WRMqRm5mwrHP4I9_wk-2KEcHSeNMOPdfXhfbeDNLyFz-NXbvuPAzrnI/s1600/picsym1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="59" data-original-width="253" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi97DsITJiu2o4Sd4yMlELRHl4o1eea9chWtHmHcs3zo3H03VNj9l4UQUUm2VE2fUhsXqPMPzEUx5Sy39ce7nC4WRMqRm5mwrHP4I9_wk-2KEcHSeNMOPdfXhfbeDNLyFz-NXbvuPAzrnI/s1600/picsym1.png" /></a></div>
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Pretty simple, right? Now, let’s see some examples.<br />
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First example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/5%5Ccdot%5Cbegin%7Bpmatrix%7D4%262%261%5C%5C%200%2613%26-2%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjM7Ywrcn-zrvuLZDfJt5l18TmHVAfBrOShJ2t8F1YDjIaBhVY-W_SXPlmDcnk__1ymfKVi-qUVnYpEymdEyvEmRzyr-nGhXuKi1czdnUcRbEKorTrElavQw-OGBmWgVQL1gfgMsnok798/s1600/picsym2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="59" data-original-width="168" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjM7Ywrcn-zrvuLZDfJt5l18TmHVAfBrOShJ2t8F1YDjIaBhVY-W_SXPlmDcnk__1ymfKVi-qUVnYpEymdEyvEmRzyr-nGhXuKi1czdnUcRbEKorTrElavQw-OGBmWgVQL1gfgMsnok798/s1600/picsym2.png" /></a></div>
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1.<span style="white-space: pre;"> </span>Multiply each of the matrix elements by the scalar<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMEhS7j-8M7PUKOf2bTrbDwlxZVE777ngxF7qRoRvcoOE0e-BE4NOH2mO3EqDgnrpVPHq0QvzhHCAzLGQabm6Xn0b05lVYw-3bGFMWFKrUN1_-nYiFFKHz_s4h7OGLfIkpviERd6dQ0YI/s1600/picsym3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="63" data-original-width="219" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMEhS7j-8M7PUKOf2bTrbDwlxZVE777ngxF7qRoRvcoOE0e-BE4NOH2mO3EqDgnrpVPHq0QvzhHCAzLGQabm6Xn0b05lVYw-3bGFMWFKrUN1_-nYiFFKHz_s4h7OGLfIkpviERd6dQ0YI/s1600/picsym3.png" /></a></div>
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2.<span style="white-space: pre;"> </span>Simplify<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2xpLPGcvewz1bWgfM-VRZ23bvbBNa1R0OgOirGj1VCkzCXCoQE8E679fAmX96BQQQSSDCcjNdSyfvDKk-p6klK-Dj6qhqMMXvxPCKqE46Q_p1dWqasFl41H-D1-1iUkKmEyR9B3b7_6Y/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="153" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2xpLPGcvewz1bWgfM-VRZ23bvbBNa1R0OgOirGj1VCkzCXCoQE8E679fAmX96BQQQSSDCcjNdSyfvDKk-p6klK-Dj6qhqMMXvxPCKqE46Q_p1dWqasFl41H-D1-1iUkKmEyR9B3b7_6Y/s1600/picsym4.png" /></a></div>
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Next example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/2%5Ccdot%5Cbegin%7Bpmatrix%7D%5Cfrac%7B1%7D%7B4%7D%26%5Cfrac%7B1%7D%7B10%7D%5C%5C%20%5Cfrac%7B2%7D%7B3%7D%26-4%5C%5C%20-%5Cfrac%7B4%7D%7B7%7D%263%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6mnNfgDj7_4e1Y1JH9ICe4OfsfaBmho_T8jU-fyJNZG8OxTTdmuFVvaBkzfU4EgZRSQapmjUjC3aLc0DK41spRB7UkVZECa3CP06fYUyEVQFvO4DftDtRrTK6z8pRRhbf6NC-IakWhXA/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="141" data-original-width="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6mnNfgDj7_4e1Y1JH9ICe4OfsfaBmho_T8jU-fyJNZG8OxTTdmuFVvaBkzfU4EgZRSQapmjUjC3aLc0DK41spRB7UkVZECa3CP06fYUyEVQFvO4DftDtRrTK6z8pRRhbf6NC-IakWhXA/s1600/picsym5.png" /></a></div>
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1.<span style="white-space: pre;"> </span>Multiply each of the matrix elements by the scalar<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyjaDJF4wxcfYBooYY-NayqWijrsg8kIL8rVeEtF8eA0Z45OXHReG5PGCo57pSS2L45-eXHjbPX8dmG6oI_ecqUk3qpnhiRpZ0o2RhcmuAvm3gBgUs-z-hymNs_KjDtXjHqpU9A68ehpU/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="141" data-original-width="175" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyjaDJF4wxcfYBooYY-NayqWijrsg8kIL8rVeEtF8eA0Z45OXHReG5PGCo57pSS2L45-eXHjbPX8dmG6oI_ecqUk3qpnhiRpZ0o2RhcmuAvm3gBgUs-z-hymNs_KjDtXjHqpU9A68ehpU/s1600/picsym6.png" /></a></div>
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2.<span style="white-space: pre;"> </span>Simplify<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjr6RuMqDPg308-hOmgLjhMDomsHMz1e-OEaYKY1tcmITStjMH0KsJ516FkzVf7zGo4av-7DL37KNezYv2hOKeLKd_lzbwgwFiZaJVCMBfV4H9HK0lbIkDBtjNmLnUdSGVn_TCUyxtlQnI/s1600/picsym7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="140" data-original-width="127" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjr6RuMqDPg308-hOmgLjhMDomsHMz1e-OEaYKY1tcmITStjMH0KsJ516FkzVf7zGo4av-7DL37KNezYv2hOKeLKd_lzbwgwFiZaJVCMBfV4H9HK0lbIkDBtjNmLnUdSGVn_TCUyxtlQnI/s1600/picsym7.png" /></a></div>
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Last example (<u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cbegin%7Bpmatrix%7Dt%5E%7B2%7D%264t%5C%5C%20-t%261%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span></u>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip2RbDIP_Ych3lK_z8mhpxGifBNtpM0CV7xPbrULCvSc1fncOsgG3yZzSV-69qFUhTr9G5pur94OHmI3y5lh5FWHWrtFFYdvtXPKPIArHp8QTV3l7olfM3TIB6Y8qLgjIPEULqzR_xl-U/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="68" data-original-width="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip2RbDIP_Ych3lK_z8mhpxGifBNtpM0CV7xPbrULCvSc1fncOsgG3yZzSV-69qFUhTr9G5pur94OHmI3y5lh5FWHWrtFFYdvtXPKPIArHp8QTV3l7olfM3TIB6Y8qLgjIPEULqzR_xl-U/s1600/picsym8.png" /></a></div>
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1.<span style="white-space: pre;"> </span>Multiply each of the matrix elements by the scalar<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXtNdj7kAVxNm6UIroQk1vCV6N-rsvIifFyMGzqSO0owixztykCkbBwnQkn-Ug4lK3NPZq28lLDbim2MzVQgt75Rn8xp1fJrpDynUJmPb0dGzGVWqHRF3z0dqVS1l3DAuIYd4wPcptsVg/s1600/picsym9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="97" data-original-width="163" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXtNdj7kAVxNm6UIroQk1vCV6N-rsvIifFyMGzqSO0owixztykCkbBwnQkn-Ug4lK3NPZq28lLDbim2MzVQgt75Rn8xp1fJrpDynUJmPb0dGzGVWqHRF3z0dqVS1l3DAuIYd4wPcptsVg/s1600/picsym9.png" /></a></div>
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2.<span style="white-space: pre;"> </span>Simplify<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwtmGnznhE6gtTziQW3f3Mp_JdYkjgAzdm2BdmIga0osokg-kn_CUu3dYWGKZFpK3aW8ZsYSI8Zjlw-cAeBZXydDlHr0xgZMJzhXejNf2i1t_Dctb9MvkdcPmGpaJ1OASw6xtt-rkp1X4/s1600/picsym10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="98" data-original-width="121" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwtmGnznhE6gtTziQW3f3Mp_JdYkjgAzdm2BdmIga0osokg-kn_CUu3dYWGKZFpK3aW8ZsYSI8Zjlw-cAeBZXydDlHr0xgZMJzhXejNf2i1t_Dctb9MvkdcPmGpaJ1OASw6xtt-rkp1X4/s1600/picsym10.png" /></a></div>
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As you can see, scalar multiplication is pretty simple. In a later blog post, we will go over matrix multiplication. For more help or practice on this topic, go to Symbolab’s <u><span lang="EN" style="color: #1155cc; font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice"><span style="color: #1155cc;">Practice</span></a></span></u>.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-84459675233858509742018-06-05T06:16:00.001-07:002022-06-08T07:56:54.230-07:00Advanced Math Solutions - Matrix Multiply, Power Calculator, Matrix PowersIf you haven’t mastered matrix multiplication, check out our <a href="http://blog.symbolab.com/2018/04/advanced-math-solutions-matrix-multiply.html" target="_blank">last blog post</a> on matrix multiplication before continuing onto the next topic. In this blog post, we will talk about powers of matrices.<br />
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The <b>power</b>, n, <b>of a matrix</b>, A, is when you multiply the matrix by itself n times. <b>A matrix can only be raised to a power if it has the same number of rows and columns</b>. Below you can visualize how to take the power of a matrix. (Note: The matrix is being multiplied by itself n times)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJpyOsSMLlXDmSbxEDlCUv00EbgqT9Zv_iAtNM8LBfVdceK00F7PW58LO7A-IaQVgJ5C45oTUnc5nFnEgKcxxb1Y4IV2qe79l2AhHgs3nil6e9Liyq1AEyFv2Zc5vEbuHrK0IpHTTkens/s1600/picsym1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="63" data-original-width="373" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJpyOsSMLlXDmSbxEDlCUv00EbgqT9Zv_iAtNM8LBfVdceK00F7PW58LO7A-IaQVgJ5C45oTUnc5nFnEgKcxxb1Y4IV2qe79l2AhHgs3nil6e9Liyq1AEyFv2Zc5vEbuHrK0IpHTTkens/s1600/picsym1.png" /></a></div>
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In order to take the power of a matrix:<br />
<br />
<ol>
<li><span style="white-space: pre;"> </span>Rewrite the problem (expand)</li>
<li><span style="white-space: pre;"> </span>Multiply the first two matrices</li>
<li><span style="white-space: pre;"> </span>Multiply the next matrix (if there is one) to the matrix produced in step 2</li>
<li><span style="white-space: pre;"> </span>Multiply the next matrix (if there is one) to the matrix produced in step 3</li>
</ol>
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And so on until you get your final matrix.<br />
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Let’s see some examples of how to take the power of a matrix to better understand..<br />
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First example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D4%262%5C%5C%202%266%5Cend%7Bpmatrix%7D%5E%7B2%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCsI2MxaRN7B-WaiB7jmScpQUGtOsN5Eg1U9VATcZnLtqcxkwkQeb4xKxE0-Ye9ezvqHvVNayPZ9vQSzr3M5mu7jF5r3agt11VVVE2N6hiTK8ORzoUtdtQoZ-OCphJZdy26UV3LsAiUXI/s1600/picsym2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="98" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCsI2MxaRN7B-WaiB7jmScpQUGtOsN5Eg1U9VATcZnLtqcxkwkQeb4xKxE0-Ye9ezvqHvVNayPZ9vQSzr3M5mu7jF5r3agt11VVVE2N6hiTK8ORzoUtdtQoZ-OCphJZdy26UV3LsAiUXI/s1600/picsym2.png" /></a></div>
<blockquote class="tr_bq">
<span style="white-space: pre;"> 1. </span>Rewrite the problem</blockquote>
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<blockquote class="tr_bq">
<span style="white-space: pre;"> 2. </span>Multiply the first two matrices</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7CvycccJYeBk1w3sfRSn1ecMphw9tGHclXjBL7ZFeZHsajc3Eckc92uwGO2BTiz40nQGgwQlcHRjJ15GAKsvMcU6yOKvbRKbZD3nf8nbwDpEjEXHhKU0Yf31NJkQsOUvd8AmctCNLECA/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="57" data-original-width="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7CvycccJYeBk1w3sfRSn1ecMphw9tGHclXjBL7ZFeZHsajc3Eckc92uwGO2BTiz40nQGgwQlcHRjJ15GAKsvMcU6yOKvbRKbZD3nf8nbwDpEjEXHhKU0Yf31NJkQsOUvd8AmctCNLECA/s1600/picsym4.png" /></a></div>
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Next problem (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D1%261%5C%5C%20%20%203%262%5Cend%7Bpmatrix%7D%5E%7B3%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<blockquote class="tr_bq">
<span style="white-space: pre;"> </span> 1. Rewrite the problem</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6aappTqC33HV-yJRBONkh8ZWyDxdvswijKUn8tkl_NGCm3NMclgSD0UWgftJ9z6CixUWeXh5zBLFZzg942zsCtB7suLQtA6PE05nzLZpspZh0VTJOcUj-eiwJSsABSIFZj05NGioHgRs/s1600/picsym7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6aappTqC33HV-yJRBONkh8ZWyDxdvswijKUn8tkl_NGCm3NMclgSD0UWgftJ9z6CixUWeXh5zBLFZzg942zsCtB7suLQtA6PE05nzLZpspZh0VTJOcUj-eiwJSsABSIFZj05NGioHgRs/s1600/picsym7.png" /></a></div>
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<blockquote class="tr_bq">
<span style="white-space: pre;"> 2. </span>Multiply the first two matrices</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiB8RO4ik4JPN1J_7OsXvXiwWNONvn9MZLSXdSNrzXfwZXoHTjr0dmqUtm5w6e-li_jPbKZGzriiEB78Ruhav0efGEjlXAbkv2KoupTJrtW6QKpIIo09zocHujW_Eb3hXYvEQo1PenA_1Q/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="159" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiB8RO4ik4JPN1J_7OsXvXiwWNONvn9MZLSXdSNrzXfwZXoHTjr0dmqUtm5w6e-li_jPbKZGzriiEB78Ruhav0efGEjlXAbkv2KoupTJrtW6QKpIIo09zocHujW_Eb3hXYvEQo1PenA_1Q/s1600/picsym8.png" /></a></div>
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<blockquote class="tr_bq">
<span style="white-space: pre;"> 3. </span>Multiply the next matrix to the matrix produced in step 2</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2ljkaX2uNoXj6QPopn3040-cKJCbLjM1Rff621RM8kYAY9FfmH8H_d5Q8varLaYGktLqyasTF6uxkjFh8i97WxJkNF5hdJ0W8EQ4XWbp0qFFrN14rlfX8e_thG40VVinDOplHgcKaGDo/s1600/picsym11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2ljkaX2uNoXj6QPopn3040-cKJCbLjM1Rff621RM8kYAY9FfmH8H_d5Q8varLaYGktLqyasTF6uxkjFh8i97WxJkNF5hdJ0W8EQ4XWbp0qFFrN14rlfX8e_thG40VVinDOplHgcKaGDo/s1600/picsym11.png" /></a></div>
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Last example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D1%260%264%5C%5C%205%262%261%5C%5C%203%262%260%5Cend%7Bpmatrix%7D%5E%7B2%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFkxATRPblVamZBPOLLHnmcu1OEQHqUJ3WNFOuhQpaxoclz1gp7WYWfwegQiJcjrNvRMruBQRqX5YwnXcuZqoAKyhgDbvYYcGhepJZbJga0EVxISi3WNL6RGDP70ONCfJOV9MbmDSvRHE/s1600/picsym14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="86" data-original-width="123" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFkxATRPblVamZBPOLLHnmcu1OEQHqUJ3WNFOuhQpaxoclz1gp7WYWfwegQiJcjrNvRMruBQRqX5YwnXcuZqoAKyhgDbvYYcGhepJZbJga0EVxISi3WNL6RGDP70ONCfJOV9MbmDSvRHE/s1600/picsym14.png" /></a></div>
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<blockquote class="tr_bq">
<span style="white-space: pre;"> 1. </span>Rewrite the problem</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMzUgwdrSXV05BRce13od3Y_hyLn8GC3Os9Q-32CWbznMWLRqNKBLn6edDfEKfpJ-bSzjk1FLhYfCOPiCVi90fYY6h0OR-zUF46Bv25sIaa1PbWu8iRYqPdHvMrya1FmzOo6EEYaRcZro/s1600/picsym15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="83" data-original-width="254" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMzUgwdrSXV05BRce13od3Y_hyLn8GC3Os9Q-32CWbznMWLRqNKBLn6edDfEKfpJ-bSzjk1FLhYfCOPiCVi90fYY6h0OR-zUF46Bv25sIaa1PbWu8iRYqPdHvMrya1FmzOo6EEYaRcZro/s1600/picsym15.png" /></a></div>
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<blockquote class="tr_bq">
<span style="white-space: pre;"> 2. </span>Multiply the two matrices</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvtv69-QnstyWje3VD8WtjmA4rcWnMOdgSjKu5bzSHQ42oIRjQoX6IUj8jcRo0ilyMkmJteDYAVxonlY_tFmq6nb1WQkJJ6FF5GHb1wepzo1Pow0uiRT0UObLFRdjBk7x6281H7t1kUac/s1600/picsym16.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="81" data-original-width="588" height="55" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvtv69-QnstyWje3VD8WtjmA4rcWnMOdgSjKu5bzSHQ42oIRjQoX6IUj8jcRo0ilyMkmJteDYAVxonlY_tFmq6nb1WQkJJ6FF5GHb1wepzo1Pow0uiRT0UObLFRdjBk7x6281H7t1kUac/s400/picsym16.png" width="400" /></a></div>
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As you can see, matrix powers aren’t complicated, as long as you’ve mastered your matrix multiplication. If you need more help with topic, check out Symbolab’s <span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice"><span style="color: #1155cc;">Practice</span></a></span>, which will provide you with practice problems and quizzes.<br />
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Until next time,<br />
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LeahUnknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-65885173853774523452018-04-24T08:56:00.001-07:002022-06-08T07:56:54.935-07:00Advanced Math Solutions - Matrix Multiply, Power Calculator, Matrix MultiplicationIn a <a href="http://blog.symbolab.com/2018/03/high-school-math-solutions-matrix_31.html" target="_blank">previous blog post</a>, we talked about one type of multiplication for matrices, scalar multiplication. In this blog post be will talk about the other type of multiplication for matrices, matrix multiplication.<br />
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Matrix multiplication is when you multiply matrix A, an n x m matrix, by matrix B, an m x p matrix, to get their product, matrix C, and n x p matrix. This means you can only multiply matrices, where matrix A has the same amount of columns as there are rows in matrix B.<br />
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In order to multiply matrices, we will have to calculate the dot product of the rows of the first matrix, matrix A, and the columns of the second matrix, matrix B.<br />
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Let’s see what this looks like:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifBTuNpVTws8VqzbvA7kvMJEcisC6D2vu9jxqHl8e_pP9i0Kn9IivnHrFZ2NZ4K6038UJISkyiuUIVL-G7XK3Hg5fdkZX7K1d2mehI7_QZW8BhrDXOeFE5385qszrRJFeKr_jIBCCxGTM/s1600/picsym1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="59" data-original-width="190" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifBTuNpVTws8VqzbvA7kvMJEcisC6D2vu9jxqHl8e_pP9i0Kn9IivnHrFZ2NZ4K6038UJISkyiuUIVL-G7XK3Hg5fdkZX7K1d2mehI7_QZW8BhrDXOeFE5385qszrRJFeKr_jIBCCxGTM/s1600/picsym1.png" /></a></div>
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In words, what we are seeing is the dot product of the first row of matrix A and the first column of matrix B make an element in the 1st row and 1st column of their product. The dot product of the second row of matrix A and the first column of matrix B make an element in the 2nd row and 1st column. The dot product of the first row of matrix A and the second column of matrix B make an element in the 1st row and 2nd column. And so on . . .<br />
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Now, let’s see some examples to help better understand how multiply matrices.<br />
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First example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D2%264%5C%5C%201%265%5Cend%7Bpmatrix%7D%5Ccdot%5Cbegin%7Bpmatrix%7D9%261%5C%5C%200%268%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDbbJavA8zU21-TMqz_jb1d3MwmzHc4wr79wPQeFq6fzWn6wuP6HyoF3e5oGv22o7md4eAmUoYZeVjAzNyRSuYZi9zCpSWHHqLe9eg17-LydfbL0aoh0i_G-HmJFo5OptHd-dzznRhbII/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="63" data-original-width="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDbbJavA8zU21-TMqz_jb1d3MwmzHc4wr79wPQeFq6fzWn6wuP6HyoF3e5oGv22o7md4eAmUoYZeVjAzNyRSuYZi9zCpSWHHqLe9eg17-LydfbL0aoh0i_G-HmJFo5OptHd-dzznRhbII/s1600/picsym4.png" /></a></div>
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<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Take the dot product of the rows of matrix A and the columns of matrix B</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5OVzqpYNu7pSc4IHhQXy42lxq-ikDVKYso-suMvX80WUX-KhzdeR5QDW1srQnFaBOMzf5vtrm4SQugzMq_SK0PUhHfpVqGIjgdwgm7QZcZjANiLU3ciMDYpADUJQydsRDb8zpzrg9tOY/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="65" data-original-width="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5OVzqpYNu7pSc4IHhQXy42lxq-ikDVKYso-suMvX80WUX-KhzdeR5QDW1srQnFaBOMzf5vtrm4SQugzMq_SK0PUhHfpVqGIjgdwgm7QZcZjANiLU3ciMDYpADUJQydsRDb8zpzrg9tOY/s1600/picsym5.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL2PffbD7DOv5z-jya1zoB4sKR1g8pEm0cQ_VQOS7V4tByT9YgM8SPfAQTkzM8Wxw8HDVeU9hYCkn2YLKMl9sDFd5_RpwvIek1f0inUpV09ZAgOjy-GyY_5FoGznKXc6QGQNPNonP0GpU/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL2PffbD7DOv5z-jya1zoB4sKR1g8pEm0cQ_VQOS7V4tByT9YgM8SPfAQTkzM8Wxw8HDVeU9hYCkn2YLKMl9sDFd5_RpwvIek1f0inUpV09ZAgOjy-GyY_5FoGznKXc6QGQNPNonP0GpU/s1600/picsym6.png" /></a></div>
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<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Simplify</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_5e64BRTou4ONGlA91QEg8fsCjHwyRkchDgTH5talo4x79ikLPggx-Tom6uxdsOAAXNJpEjVNHx3-8Bl3fh3RqA462V5_HVm-_kX9JSrR2zb1mM4ngjaCoDfzj8I0zHSlCCxzL0Nehww/s1600/picsym7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="109" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_5e64BRTou4ONGlA91QEg8fsCjHwyRkchDgTH5talo4x79ikLPggx-Tom6uxdsOAAXNJpEjVNHx3-8Bl3fh3RqA462V5_HVm-_kX9JSrR2zb1mM4ngjaCoDfzj8I0zHSlCCxzL0Nehww/s1600/picsym7.png" /></a></div>
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Next example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D1%265%262%5Cend%7Bpmatrix%7D%5Ccdot%5Cbegin%7Bpmatrix%7D4%268%2610%5C%5C%200%261%262%5C%5C%207%2611%260%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmimOM512iwtzlpDijeFzVMb2WnCbDqQr95yrwYri6I4CgpaMfYClP3cFbOOy26Z_CuW0UXQqCWGOzvINn5UxysHthclbVkLOVPH6GQvpv-uwJtc3btL3KifbXe-UaacqzrAMplkzPlKM/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="83" data-original-width="245" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmimOM512iwtzlpDijeFzVMb2WnCbDqQr95yrwYri6I4CgpaMfYClP3cFbOOy26Z_CuW0UXQqCWGOzvINn5UxysHthclbVkLOVPH6GQvpv-uwJtc3btL3KifbXe-UaacqzrAMplkzPlKM/s1600/picsym8.png" /></a></div>
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<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Take the dot product of the rows of matrix A and the columns of matrix B</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgP9lDscsAQg-44nEUD0FgUgMtDmoC3Ot_Lbuih1hLArfmS_PdgotQQdAHPEEDMmG6vEOGh-m1kB6NixNNX3O9-ww3TsfO02lxQ3rZUOqZ5pPRKEK1YO23tvB9xqvAX-TgCuv4NC5opNOo/s1600/picsym9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="40" data-original-width="512" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgP9lDscsAQg-44nEUD0FgUgMtDmoC3Ot_Lbuih1hLArfmS_PdgotQQdAHPEEDMmG6vEOGh-m1kB6NixNNX3O9-ww3TsfO02lxQ3rZUOqZ5pPRKEK1YO23tvB9xqvAX-TgCuv4NC5opNOo/s1600/picsym9.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbZT0rUFDjKgkhDq44jh89fLWh_sclPFdnldyRzhWmCB935qUxukmjKVyY-Rug8j65SsH0Bw6TdfEUDncCNYkkdkHr8XWoagGdczNBVYRVY2eCK0yqkIHCEDLhMq-z4H2ZdLgv3d4yfvY/s1600/picsym10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="37" data-original-width="370" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbZT0rUFDjKgkhDq44jh89fLWh_sclPFdnldyRzhWmCB935qUxukmjKVyY-Rug8j65SsH0Bw6TdfEUDncCNYkkdkHr8XWoagGdczNBVYRVY2eCK0yqkIHCEDLhMq-z4H2ZdLgv3d4yfvY/s1600/picsym10.png" /></a></div>
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<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Simplify</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbEDP_rMGaNs2nnS3V6eXT9pWXH89ooQu8wBHsuMPyg8H7o2QBC3VogjXox58yYKsVhsQifCUD_6NI_Oq9_trpwJKWNFH_ym6HHpq4AkJ5ZRA4ahUres8XCZ-ZxIbatBlhW2ZB9Lo36JM/s1600/picsym11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="36" data-original-width="157" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbEDP_rMGaNs2nnS3V6eXT9pWXH89ooQu8wBHsuMPyg8H7o2QBC3VogjXox58yYKsVhsQifCUD_6NI_Oq9_trpwJKWNFH_ym6HHpq4AkJ5ZRA4ahUres8XCZ-ZxIbatBlhW2ZB9Lo36JM/s1600/picsym11.png" /></a></div>
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Last example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D11%20%26%203%20%5C%5C7%20%26%2011%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D8%20%26%200%20%26%201%20%5C%5C0%20%26%203%20%26%205%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYZjaaxAjpz3MwAQ3paD0_dJGjPS2NDivnnuEPMS0Hohy8nFWAUpfspg4NLFj3WqW5HsHueu0MTvk9I3cLkMB09Ni4PPDnfw7ZvEWqKgJxUdsDW9qpcczkIBTQCmPcX7mD9LkP71Wg-dM/s1600/picsym12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="60" data-original-width="226" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYZjaaxAjpz3MwAQ3paD0_dJGjPS2NDivnnuEPMS0Hohy8nFWAUpfspg4NLFj3WqW5HsHueu0MTvk9I3cLkMB09Ni4PPDnfw7ZvEWqKgJxUdsDW9qpcczkIBTQCmPcX7mD9LkP71Wg-dM/s1600/picsym12.png" /></a></div>
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<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Take the dot product of the rows of matrix A and the columns of matrix B</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-kNZabhVT5X2S5GTU-suio2HOrP-XAFkU3qxSvm-lcZ8okvdHzdnwUVKj7r4GWRX6ydVvxdzMVLdWp14pMHGr78KVL4IRclW4PTzaaSgTkUUj8fdOk5wov3kaSE3mEPOYEkP3eHHdPPE/s1600/picsym13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="507" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-kNZabhVT5X2S5GTU-suio2HOrP-XAFkU3qxSvm-lcZ8okvdHzdnwUVKj7r4GWRX6ydVvxdzMVLdWp14pMHGr78KVL4IRclW4PTzaaSgTkUUj8fdOk5wov3kaSE3mEPOYEkP3eHHdPPE/s1600/picsym13.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1KPl-VldkHGrnW-eJSXvlpSuFvktLRa4LY7ABShTVKc96v8MxoD3FuKP_O_zImWVwPV_irnHcnRq3m-58CBLKx11OGYZs2Ut7zjd5YHGd4Bb60g5ZlNn9xl-Aa7-9vmu_xduzG8tJCAw/s1600/picsym14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="289" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1KPl-VldkHGrnW-eJSXvlpSuFvktLRa4LY7ABShTVKc96v8MxoD3FuKP_O_zImWVwPV_irnHcnRq3m-58CBLKx11OGYZs2Ut7zjd5YHGd4Bb60g5ZlNn9xl-Aa7-9vmu_xduzG8tJCAw/s1600/picsym14.png" /></a></div>
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<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Simplify</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOtRaDyyVM1MlNvTR5I7JLX63MqW4HyuqFiMxAHzeWw9TwGHi-esopVh70P9Jty3k3GnhDpkadJubZy4-Z0Dv4SejCZ56ANkgWPb1uT3K6ZjA2pjyHhhLB3_OBpxjo3h1ASu4t8AnV6dQ/s1600/picsym15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="59" data-original-width="181" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOtRaDyyVM1MlNvTR5I7JLX63MqW4HyuqFiMxAHzeWw9TwGHi-esopVh70P9Jty3k3GnhDpkadJubZy4-Z0Dv4SejCZ56ANkgWPb1uT3K6ZjA2pjyHhhLB3_OBpxjo3h1ASu4t8AnV6dQ/s1600/picsym15.png" /></a></div>
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Matrix multiplication can be difficult and tricky, when learning it for the first time. The more you practice it, the more it’ll become second nature to you. Check out Symbolab’s <span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice/matrices-practice?subTopic=Multiply"><span style="color: #1155cc;">Practice</span></a></span> for practice problems and quizzes on this topic.<br />
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Until next time,<br />
<br />
Leah<br />
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Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-54461691634796808242018-03-31T08:44:00.001-07:002022-06-08T07:56:55.821-07:00High School Math Solutions - Matrix Transpose Calculator, TransposeThe transpose of a matrix is when you turn all the rows of a matrix into columns and vice versa. Row 1 becomes column 1, row 2 becomes column 2, and so on. The transpose of a matrix, A, is denoted <span class="mathquill-embedded-latex font16">A^T</span>.<br />
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When you transpose a matrix, the element in row i, column j becomes the element in row j, column i of the transposed matrix.<br />
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Let’s see some examples to better understand what the transpose of a matrix is.<br />
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First example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/transpose%20%5Cbegin%7Bpmatrix%7D1%26-2%5C%5C%200%262%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcj_9kjTZaP30aYraoV6udHavneMTro5ZE48RdzmutorZ2Nxmodv9Oi1CBt__pJCEraHRwFL3ksRm7qjd8Afp_AjzX1m06H4c8RhnldFtKhrMVuPWgH0QfXclptklr9ZKWlfh3CSbWB_o/s1600/picsym11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="191" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcj_9kjTZaP30aYraoV6udHavneMTro5ZE48RdzmutorZ2Nxmodv9Oi1CBt__pJCEraHRwFL3ksRm7qjd8Afp_AjzX1m06H4c8RhnldFtKhrMVuPWgH0QfXclptklr9ZKWlfh3CSbWB_o/s1600/picsym11.png" /></a></div>
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<span style="white-space: pre;"> 1. </span>Turn the rows into columns<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqUjBKCuC6F1YyVm229jaOBLq8vjuFMBKumUmoD9spCLp5mn609payoNMhhyphenhyphen2zeBYeNL5F1U5hGE-zUd0l8LhgrKFeXwbxe6pTQkDxIpH04ys2cIVbuLUQkZ-jYNeYrMmOi7i1GPffLwU/s1600/picsym12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="61" data-original-width="96" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqUjBKCuC6F1YyVm229jaOBLq8vjuFMBKumUmoD9spCLp5mn609payoNMhhyphenhyphen2zeBYeNL5F1U5hGE-zUd0l8LhgrKFeXwbxe6pTQkDxIpH04ys2cIVbuLUQkZ-jYNeYrMmOi7i1GPffLwU/s1600/picsym12.png" /></a></div>
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Next example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D5%262%263%5C%5C%205%26-1%26-7%5C%5C%2013%266%2627%5Cend%7Bpmatrix%7D%5E%7BT%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6jIV_Qin_uZFv7fwTsQ9bfePyLQ5k9MM02wNAvecAPoEyfqiT9A4KQg1HdBWqn-NIz6lgEn2a01Kv7GcD7bm-ZVV59VjqdwfxE5FOflLByQvY1U2y6PXLCper-pIZO0_y-BFaWEDuqAo/s1600/picsym13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="86" data-original-width="167" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6jIV_Qin_uZFv7fwTsQ9bfePyLQ5k9MM02wNAvecAPoEyfqiT9A4KQg1HdBWqn-NIz6lgEn2a01Kv7GcD7bm-ZVV59VjqdwfxE5FOflLByQvY1U2y6PXLCper-pIZO0_y-BFaWEDuqAo/s1600/picsym13.png" /></a></div>
<br />
<span style="white-space: pre;"> 1. </span>Turn the rows into columns<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitqcfuTze6DH1F44KmUGBKa5DtQr3UerLBrnXL4je8z0dzrRJqcR6lJJUGXxUpeMasQdoeIeh4gcTzs4GV2AsW5mZWwEhTSPqwmod5mgQv2nHVdX4RbQjPo9XxDp1x5sE3p23p9nY08rg/s1600/picsym14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="79" data-original-width="132" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitqcfuTze6DH1F44KmUGBKa5DtQr3UerLBrnXL4je8z0dzrRJqcR6lJJUGXxUpeMasQdoeIeh4gcTzs4GV2AsW5mZWwEhTSPqwmod5mgQv2nHVdX4RbQjPo9XxDp1x5sE3p23p9nY08rg/s1600/picsym14.png" /></a></div>
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Last example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/transpose%5Cbegin%7Bpmatrix%7D5%267%5C%5C%204%261%5C%5C%20-1%266%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<span style="white-space: pre;"> 1. </span>Turn the rows into columns<br />
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This concept can be hard to visualize, so practicing a few examples will help you understand and become familiar with the transpose of a matrix. For more help or practice on the transpose of a matrix and other related matrix topics, visit Symbolab’s <span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice/"><span style="color: #1155cc;">Practice</span></a></span>.<br />
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Until next time,<br />
<br />
Leah<br />
<br />Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-38543846358375364122018-03-21T08:46:00.001-07:002022-06-08T07:56:56.969-07:00High School Math Solutions - Matrix Multiply Calculator, Matrix Scalar MultiplicationLast blog post, we talked about how to add and subtract matrices. Now, we will start getting into multiplication for matrices. There are two types: scalar multiplication and matrix multiplication. In this blog post, we will talk about the simpler of the two, scalar multiplication.<br />
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<b>Scalar multiplication</b> is when you multiply a matrix by a value, called a <b>scalar</b>. In scalar multiplication, you multiply each element of the matrix by the scalar.<br />
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Here is what scalar multiplication looks like:<br />
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Pretty simple, right? Now, let’s see some examples.<br />
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First example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/5%5Ccdot%5Cbegin%7Bpmatrix%7D4%262%261%5C%5C%200%2613%26-2%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip2HjBNGE64928w6a6dF9LXQqpQ5U8BX3IGfHXMgzMcSlyTgJFPkU8tCapXpW6HRmiHveOakcB0J65OopZTNSkTHWIspWo6KdYg2fhLkLuRjMFgyCEenVe_BXC0nLullP3ki26DxBhYCw/s1600/picsym2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="59" data-original-width="168" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip2HjBNGE64928w6a6dF9LXQqpQ5U8BX3IGfHXMgzMcSlyTgJFPkU8tCapXpW6HRmiHveOakcB0J65OopZTNSkTHWIspWo6KdYg2fhLkLuRjMFgyCEenVe_BXC0nLullP3ki26DxBhYCw/s1600/picsym2.png" /></a></div>
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<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Multiply each of the matrix elements by the scalar</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6Ljt90057XrIMQCbLNw0ESVNkQlrBNGNr1Wp58akHDQ-kF9h6xCsoJQQv_qZkvRSawLjrAv0pMup_eCybgmgKJpsp285A_uvr44qNTzAhyphenhyphen8sLmFDiRCMGPfQbzW9RSKG_izJn7hLSt2M/s1600/picsym3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="63" data-original-width="219" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6Ljt90057XrIMQCbLNw0ESVNkQlrBNGNr1Wp58akHDQ-kF9h6xCsoJQQv_qZkvRSawLjrAv0pMup_eCybgmgKJpsp285A_uvr44qNTzAhyphenhyphen8sLmFDiRCMGPfQbzW9RSKG_izJn7hLSt2M/s1600/picsym3.png" /></a></div>
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<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Simplify</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjth15_wZMx3Ggf1OoDr7k7zuWk2zhL414J5neCdDdSTJ4QdMqHkQhnySNW0qHWjsy07Cd0jPBKZtkeDZG-YvH0LlnSrTiF-j1Mu4ycDg7RMVlN0N2ycOk4entb2Q0gz8nUTHzeJYS_2sk/s1600/picsym4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="64" data-original-width="153" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjth15_wZMx3Ggf1OoDr7k7zuWk2zhL414J5neCdDdSTJ4QdMqHkQhnySNW0qHWjsy07Cd0jPBKZtkeDZG-YvH0LlnSrTiF-j1Mu4ycDg7RMVlN0N2ycOk4entb2Q0gz8nUTHzeJYS_2sk/s1600/picsym4.png" /></a></div>
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Next example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/2%5Ccdot%5Cbegin%7Bpmatrix%7D%5Cfrac%7B1%7D%7B4%7D%26%5Cfrac%7B1%7D%7B10%7D%5C%5C%20%5Cfrac%7B2%7D%7B3%7D%26-4%5C%5C%20-%5Cfrac%7B4%7D%7B7%7D%263%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyWyjKsIA86q44Xj6QZHPmVJz0DTTmsYyyjCY5UDCA7GINvqMThXi0csFhv7_rc_B-dJT_m0D7BH2b7a0FC1q9QsU7HRuz_94_P6l623CjXhch8RLytO9ZzFZ1EKu5QKUbZKF_MeLjEwI/s1600/picsym5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="141" data-original-width="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyWyjKsIA86q44Xj6QZHPmVJz0DTTmsYyyjCY5UDCA7GINvqMThXi0csFhv7_rc_B-dJT_m0D7BH2b7a0FC1q9QsU7HRuz_94_P6l623CjXhch8RLytO9ZzFZ1EKu5QKUbZKF_MeLjEwI/s1600/picsym5.png" /></a></div>
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<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Multiply each of the matrix elements by the scalar</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3ri5iIOYczfbgyRw7nUchdfEIIKw5U9MJErAchEHn98KDnmApwDQCLyaoIVMMJ387YZKKQ_62aF222hl6XLDcSOmsXPt9D4Uj_f5iOoWQmDtLyOMQSG6R7HFl5q4_G7LbxUEC9oimV5s/s1600/picsym6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="141" data-original-width="175" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3ri5iIOYczfbgyRw7nUchdfEIIKw5U9MJErAchEHn98KDnmApwDQCLyaoIVMMJ387YZKKQ_62aF222hl6XLDcSOmsXPt9D4Uj_f5iOoWQmDtLyOMQSG6R7HFl5q4_G7LbxUEC9oimV5s/s1600/picsym6.png" /></a></div>
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<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Simplify</blockquote>
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Last example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cbegin%7Bpmatrix%7Dt%5E%7B2%7D%264t%5C%5C%20-t%261%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp8tIKqQgN4c685AApZ5EWep3DFxwCwhF4pF6AYQrE1mKEsCy3u6YgoO-5hlqVkRqKlFjoLhKiR0PFMzymW_wlp1gPZST-eOQZbVpnjciXZNu-i6yOBJPC9OkdvAl8wzkDJCKz_ELyRPc/s1600/picsym8.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="68" data-original-width="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp8tIKqQgN4c685AApZ5EWep3DFxwCwhF4pF6AYQrE1mKEsCy3u6YgoO-5hlqVkRqKlFjoLhKiR0PFMzymW_wlp1gPZST-eOQZbVpnjciXZNu-i6yOBJPC9OkdvAl8wzkDJCKz_ELyRPc/s1600/picsym8.png" /></a></div>
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<blockquote class="tr_bq">
1.<span style="white-space: pre;"> </span>Multiply each of the matrix elements by the scalar</blockquote>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2WXIBPM4tzetPErJ13kPJJgLwAcm3e3UDmr2IZ3JjlEu_KK2pWywAEfvOnoAS8aRwiEKxoCKb-ibg9gRclw6Fu9yNqBIKu2JlPm2BrHcwDui0cy8zQ_-egpOy_GeRRnJau4O2PekhuPU/s1600/picsym9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="97" data-original-width="163" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2WXIBPM4tzetPErJ13kPJJgLwAcm3e3UDmr2IZ3JjlEu_KK2pWywAEfvOnoAS8aRwiEKxoCKb-ibg9gRclw6Fu9yNqBIKu2JlPm2BrHcwDui0cy8zQ_-egpOy_GeRRnJau4O2PekhuPU/s1600/picsym9.png" /></a></div>
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<blockquote class="tr_bq">
2.<span style="white-space: pre;"> </span>Simplify</blockquote>
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As you can see, scalar multiplication is pretty simple. In a later blog post, we will go over matrix multiplication. For more help or practice on this topic, go to Symbolab’s <span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice"><span style="color: #1155cc;">Practice</span></a></span>.<br />
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Until next time,<br />
<br />
Leah<br />
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Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-54730797019068100422018-03-11T05:46:00.001-07:002022-06-08T07:56:58.098-07:00High School Math Solutions - Matrix Add, Subtract Calculator, MatricesA matrix is an array of numbers, symbols, or expressions that are displayed in rows and columns. The dimension of a matrix are written as r x n, where r is the number of rows and n is the number of columns. One thing you can do with matrices is add or subtract them together. The only caveat to adding or subtracting matrices is that the matrices must be the same size, i.e. must have the same dimension.<br />
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Adding and subtracting matrices is simple. Simply, add or subtract each element in the matching position, creating a new matrix with the same dimension. Let’s visualize this to get a better understanding.<br />
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Here is how to add matrices together:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUJxxP0r8ROv2qi0xpwYeSF31B3E9LWuX_6d2-mVd1SapE3NmVgJu9ummRbdX9nNkt-npVLjDqmgNTeOy91PXwrfvM6xAnQvF-wUL5iJJJbHGFmvce9YV1sXWVXUTzuq_V375xNrKUJVM/s1600/picsym34.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="60" data-original-width="327" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUJxxP0r8ROv2qi0xpwYeSF31B3E9LWuX_6d2-mVd1SapE3NmVgJu9ummRbdX9nNkt-npVLjDqmgNTeOy91PXwrfvM6xAnQvF-wUL5iJJJbHGFmvce9YV1sXWVXUTzuq_V375xNrKUJVM/s1600/picsym34.png" /></a></div>
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You’d follow the same method for subtraction.<br />
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Here is an example of matrices that can’t be added together because they aren’t the same size:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4VD_H7DBOjMkfyZTfeLmDNognKyzkm7lgpDHgKjIlnryfoma8xN4ZkDXwc0L_2sL60NTmP3wOFLFVzFepu5hjjpzNdpgdtsJaIhgwjePLVzaC7MWjz4qSMMXt-AYBHNvYNF5Og64VzS4/s1600/picsym35.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="62" data-original-width="202" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4VD_H7DBOjMkfyZTfeLmDNognKyzkm7lgpDHgKjIlnryfoma8xN4ZkDXwc0L_2sL60NTmP3wOFLFVzFepu5hjjpzNdpgdtsJaIhgwjePLVzaC7MWjz4qSMMXt-AYBHNvYNF5Og64VzS4/s1600/picsym35.png" /></a></div>
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You can see one matrix is 2x2 and the other is 2x3.<br />
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Let’s see some examples to better understand.<br />
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Here’s the first example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D4%266%262%5Cend%7Bpmatrix%7D%2B%5Cbegin%7Bpmatrix%7D2%268%264%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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1.<span style="white-space: pre;"> </span>Add elements in the matching positions<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiAIWiXE-o1D3FE-bOgUgugzrdKxrKS-1um3BjUxbCQEz0XUUZ4P-iOb7BSAhHEs4RrFhSrdQYwvQ6QTBg67ipSU24YS5gv-QkuWJNGev7vuPIMYnIoVp5pWVK-q5Tjpf51bk-E4B2_GM/s1600/picsym37.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="41" data-original-width="227" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiAIWiXE-o1D3FE-bOgUgugzrdKxrKS-1um3BjUxbCQEz0XUUZ4P-iOb7BSAhHEs4RrFhSrdQYwvQ6QTBg67ipSU24YS5gv-QkuWJNGev7vuPIMYnIoVp5pWVK-q5Tjpf51bk-E4B2_GM/s1600/picsym37.png" /></a></div>
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2. Simplify<br />
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Next example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D4%26-2%5C%5C%2011%266%5C%5C%209%268%5Cend%7Bpmatrix%7D-%5Cbegin%7Bpmatrix%7D-4%266%5C%5C%203%2615%5C%5C%208%2613%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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1.<span style="white-space: pre;"> </span>Add elements in the matching positions<br />
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2.<span style="white-space: pre;"> </span>Simplify<br />
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Last example (<span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/solver/step-by-step/%5Cbegin%7Bpmatrix%7D12%268%2615%5C%5C%20-8%2622%263%5C%5C%202%260%267%5Cend%7Bpmatrix%7D-%5Cbegin%7Bpmatrix%7D-5%261%2616%5C%5C%2028%26-10%263%5C%5C%206%264%2611%5Cend%7Bpmatrix%7D?or=blog"><span style="color: #1155cc;">click here</span></a></span>):<br />
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1.<span style="white-space: pre;"> </span>Add elements in the matching positions<br />
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2.<span style="white-space: pre;"> </span>Simplify<br />
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As you can see, adding and subtracting matrices is pretty simple. If you are interested in practicing more problems on this topic or want more help on this topic, check out Symbolab’s <span lang="EN" style="font-family: "arial" , sans-serif; font-size: 11.0pt; line-height: 115%;"><a href="https://www.symbolab.com/practice/matrices-practice?subTopic=Add/Subtract"><span style="color: #1155cc;">Practice</span></a></span>.<br />
<br />
Until next time,<br />
<br />
LeahUnknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-76604908611473733292018-01-16T10:23:00.001-08:002022-06-08T07:56:59.228-07:00Middle School Math Solutions – Order of Operations CalculatorGiven a problem like, <span class="mathquill-embedded-latex font16">3×5-2</span>, you might be wondering “how do I solve this?” since there are two possible ways.<br />
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<span style="white-space: pre;"> </span>Option 1: <span class="mathquill-embedded-latex font16">(3×5)-2=15-2=13</span><br />
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<span style="white-space: pre;"> </span>Option 2: <span class="mathquill-embedded-latex font16">3×(5-2)=3×3=9</span><br />
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This is where the order of operations comes in, since there is only one correct answer.<br />
<br />
The order of operations are rules that tell us which operation to perform first when calculating an algebraic expression.<br />
<br />
Here is the order of operations:<br />
<br />
<span style="white-space: pre;"> </span>1. Parentheses<br />
<span style="white-space: pre;"> </span>2. Exponents<br />
<span style="white-space: pre;"> </span>3. Multiplication and Division (from left to right)<br />
<span style="white-space: pre;"> </span>4. Addition and Subtraction (from left to right)<br />
<br />
When calculating an algebraic expression, go down the order of operations, starting with parentheses, and perform the operations that apply.<br />
<br />
There is an acronym used to remember the order of operations, PEMDAS (parentheses, exponents, multiplication, division, subtraction, and addition). The phrase “Please Excuse My Dear Aunt Sally” is also used to help remember the order by using the first letter of each word.<br />
<br />
Looking back at the problem above and knowing the order of operations, we now can see that the answer is 13, option 1.<br />
<br />
Let’s see some examples using the order of operations.<br />
<br />
First example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/8%5Ccdot5-%5Cleft(6%2B10%5Cright)%5Cdiv2?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">8×5-(6+10)÷2</span><br />
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<span style="white-space: pre;"> 1. </span>Calculate within parentheses<br />
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<span class="mathquill-embedded-latex font16">=8×5-16÷2</span><br />
<blockquote class="tr_bq">
We simplified what was in the parentheses.</blockquote>
<span style="white-space: pre;"> 2. </span>Multiplication and Division (from left to right)<br />
<br />
<span class="mathquill-embedded-latex font16">=40-16÷2</span><br />
<span class="mathquill-embedded-latex font16">=40-8</span><br />
<blockquote class="tr_bq">
We multiplied and divided starting from the left of the expression, making our way to the right.</blockquote>
<span style="white-space: pre;"> 3. </span>Addition and Subtraction (from left to right)<br />
<br />
<span class="mathquill-embedded-latex font16">=40-8</span><br />
<span class="mathquill-embedded-latex font16">=32</span><br />
<br />
Next example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/2%5Ccdot%5Cleft(5%5Ccdot6%2B2%5E%7B2%7D%5Cright)%2B%5Cleft(8%5Cdiv2%5Cright)?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">2×(5×6+2^2 )+(8÷2)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Calculate within parentheses – Evaluate <span class="mathquill-embedded-latex font16">(5×6+2^2)</span><br />
<br />
<span class="mathquill-embedded-latex font16">(5×6+2^2 )=(5×6+4)</span><br />
<span class="mathquill-embedded-latex font16">=(30+4)</span><br />
<span class="mathquill-embedded-latex font16">=34</span><br />
<blockquote class="tr_bq">
Make sure you apply the order of operations when calculating inside the parentheses</blockquote>
<blockquote class="tr_bq">
Now we have: <span class="mathquill-embedded-latex font16">2×(5×6+2^2 )+(8÷2)=2×34+(8÷2)</span></blockquote>
<span style="white-space: pre;"> 2. </span>Calculate within parentheses – Evaluate <span class="mathquill-embedded-latex font16">(8÷2)</span><br />
<br />
<span class="mathquill-embedded-latex font16">8÷2=4</span><br />
<blockquote class="tr_bq">
<span style="white-space: pre;"> </span>Now we have: <span class="mathquill-embedded-latex font16">2×34+(8÷2)=2×34+4</span></blockquote>
<span style="white-space: pre;"> 3. </span>Multiplication and Division (from left to right)<br />
<br />
<span class="mathquill-embedded-latex font16">=2×34+4</span><br />
<span class="mathquill-embedded-latex font16">=68+4</span><br />
<br />
<span style="white-space: pre;"> 4. </span>Addition and Subtraction (from left to right)<br />
<br />
<span class="mathquill-embedded-latex font16">=68+4</span><br />
<span class="mathquill-embedded-latex font16">=72</span><br />
<br />
Last example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/6%2B%5Cleft(2%5Cleft(4%2B2%5Cright)%5Cright)%5E%7B2%7D-5%5E%7B2%7D?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">6+(2(4+2))^2-5^2</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Calculate within parentheses<br />
<br />
<span class="mathquill-embedded-latex font16">=6+(2(6) )^2-5^2</span><br />
<span class="mathquill-embedded-latex font16">=6+12^2-5^2</span><br />
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<span style="white-space: pre;"> 2. </span>Exponents<br />
<br />
<span class="mathquill-embedded-latex font16">=6+144-5^2</span><br />
<span class="mathquill-embedded-latex font16">=6+144-25</span><br />
<br />
<span style="white-space: pre;"> 3. </span>Addition and Subtraction (from left to right)<br />
<br />
<span class="mathquill-embedded-latex font16">=150-25</span><br />
<span class="mathquill-embedded-latex font16">=125</span><br />
<br />
Learning the order of operations is essential in math because they will guide you to the correct answer. For more help or practice with the order of operations, check out Symbolab’s <span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/practice">Practice</a></span>.<br />
<br />
Until next time,<br />
<br />
LeahUnknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-82256344096408077432018-01-03T05:09:00.003-08:002022-06-08T07:56:59.873-07:00Middle School Math Solutions – Expand Calculator, Binomial ExpansionWe’ve learned how to expand <a href="http://blog.symbolab.com/2017/11/middle-school-math-solutions-expand_15.html" target="_blank">perfect squares</a> and <a href="http://blog.symbolab.com/2017/12/middle-school-math-solutions-expand_20.html" target="_blank">perfect cubes</a>. Now, we are going to learn how to expand binomials raised to any positive integers.<br />
<br />
Imagine having to expand a binomial raised to a power of 7. Sounds like a lot of work, right? Good news, we have a formula!<br />
<br />
Here is the Binomial Theorem:<br />
<br />
<span class="mathquill-embedded-latex font16">(a+b)^n=\sum_{i=0}^n\binom{n}{i}a^(n-i)b^i</span><br />
<br />
<span class="mathquill-embedded-latex font16">\binom{n}{i}</span> is a combination, which we read as “n choose i”.<br />
Here is the formula for n choose i:<br />
<br />
<span class="mathquill-embedded-latex font16">\binom{n}{i}=\frac{n!}{i!(n-i)!}</span><br />
<br />
Let’s see some examples using this formula.<br />
<br />
First example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(x%2B2%5Cright)%5E%7B%204%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(x+2)^4</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=x</span>, <span class="mathquill-embedded-latex font16">b=2</span>, <span class="mathquill-embedded-latex font16">n=4</span><br />
<br />
<span class="mathquill-embedded-latex font16">(x+2)^4=\sum_{i=0}^4\binom{4}{i}x^(4-i)2^i</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Expand the summation and simplify<br />
<br />
<span class="mathquill-embedded-latex font16">\sum_{i=0}^4\binom{4}{i}x^(4-i)2^i =\binom{4}{0} x^4∙2^0+\binom{4}{1} x^3∙2^1+\binom{4}{2} x^2∙2^2+\binom{4}{3} x^1 ∙2^3+\binom{4}{4} x^0∙2^4</span><br />
<br />
<span class="mathquill-embedded-latex font16">=\frac{4!}{0!(4-0)!} x^4∙2^0+\frac{4!}{1!(4-1)!} x^3∙2^1</span> <br />
<br />
<span class="mathquill-embedded-latex font16">+\frac{4!}{2!(4-2)!} x^2∙2^2+\frac{4!}{3!(4-3)!} x^1 ∙2^3+\frac{4!}{4!(4-4)!} x^0∙2^4</span><br />
<br />
<span class="mathquill-embedded-latex font16">=\frac{24}{(1)24} x^4∙2^0+\frac{24}{1(6)} x^3∙2^1+\frac{24}{2(2)}x^2∙2^2+\frac{24}{6(1)} x^1 ∙2^3+\frac{24}{24(1)} x^0∙2^4</span><br />
<br />
<span class="mathquill-embedded-latex font16">=x^4+4x^3∙2+6x^2∙4+4x∙8+1∙16</span><br />
<br />
<span class="mathquill-embedded-latex font16">=x^4+8x^3+24x^2+32x+16</span> <br />
<br />
You can see that there is a lot to calculate, but that this formula makes expanding easier and faster. In this example, I went into detail on how to simplify this expansion, specifically the combination. Now, that you’ve seen and understand how to calculate combinations, the next examples won’t be in such detail (you can also check out a more detailed step by step solution by clicking the hyperlinks).<br />
<br />
Next example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="file:///C:/Users/lali/Downloads/=4!/0!(4-0)!%20x%5e4%E2%88%992%5e0+4!/1!(4-1)!%20x%5e3%E2%88%992%5e1">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(x-y)^5</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=x</span>, <span class="mathquill-embedded-latex font16">b=-y</span>, <span class="mathquill-embedded-latex font16">n=5</span><br />
<br />
<span class="mathquill-embedded-latex font16">(x-y)^5=\sum_{i=0}^5\binom{5}{i} x^(5-i) (-y)^i</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Expand the summation and simplify<br />
<br />
<span class="mathquill-embedded-latex font16">\sum_{i=0}^5\binom{5}{i} x^(5-i) (-y)^i</span><br />
<span class="mathquill-embedded-latex font16">=\binom{5}{0} x^5∙(-y)^0+\binom{5}{1} x^4∙(-y)^1+\binom{5}{2} x^3∙(-y)^2+\binom{5}{3} x^2 ∙(-y)^3+\binom{5}{4} x^1∙(-y)^4+\binom{5}{5} x^0 (-y)^5</span><br />
<span class="mathquill-embedded-latex font16">=x^5-5x^4 y+10x^3 y^2-10x^2 y^3+5xy^4-y^5</span><br />
<br />
Last example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(3%2Bx%5E%7B2%7D%5Cright)%5E%7B%20%7D%5E%7B4%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(3+x^2 )^4</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=3</span>, <span class="mathquill-embedded-latex font16">b=x^2</span>, <span class="mathquill-embedded-latex font16">n=4</span><br />
<br />
<span class="mathquill-embedded-latex font16">(3+x^2)^4=\sum_{i=0}^4\binom{4}{i} 3^(4-i) (x^2)^i</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Expand the summation and simplify<br />
<br />
<span class="mathquill-embedded-latex font16">\sum_{i=0}^4\binom{4}{i} 3^(4-i) (x^2 )^i</span><br />
<span class="mathquill-embedded-latex font16">=\binom{4}{0} 3^4∙(x^2)^0+\binom{4}{1} 3^3∙(x^2 )^1+\binom{4}{2} 3^2∙(x^2)^2+\binom{4}{3} 3^1 ∙(x^2 )^3+\binom{4}{4} 3^0∙(x^2 )^4</span><br />
<br />
<span class="mathquill-embedded-latex font16">=81+4∙27x^2+6∙9x^4+4∙3x^6+x^8</span><br />
<br />
<span class="mathquill-embedded-latex font16">=81+108x^2+54x^4+12x^6+x^8</span><br />
<br />
Binomial expansions require practice to get the hang of things and to help memorize the formula. If you are interested in more practice problems on this topic or help, check out Symbolab’s <span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/practice">Practice</a></span>.<br />
<br />
Until next time,<br />
<br />
LeahUnknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-20481511018956748002017-12-20T23:33:00.001-08:002022-06-08T07:57:00.954-07:00Middle School Math Solutions – Expand Calculator, Perfect CubeSimilarly to the perfect square formula, which we covered in the <a href="http://blog.symbolab.com/2017/11/middle-school-math-solutions-expand_15.html" target="_blank">last post</a>, we get the perfect cube formula. The perfect cube is when we cube a binomial. We use polynomial multiplication and the FOIL method for this formula.<br />
<br />
Let’s take a look at it:<br />
<br />
<span class="mathquill-embedded-latex font16">(a+b)^3=a^3+3a^2 b+3ab^2+b^3</span><br />
<br />
Here’s the proof:<br />
<br />
<span class="mathquill-embedded-latex font16">(a+b)^3=(a+b)(a+b)(a+b)</span><br />
<span class="mathquill-embedded-latex font16">=(a+b)(a^2+2ab+b^2)</span><br />
<span class="mathquill-embedded-latex font16">=a(a^2+2ab+b^2 )+b(a^2+2ab+b^2)</span><br />
<span class="mathquill-embedded-latex font16">=a^3+2a^2 b+ab^2+a^2 b+2ab^2+b^3</span><br />
<span class="mathquill-embedded-latex font16">=a^3+3a^2 b+3ab^2+b^3</span><br />
<br />
As you can see there are many steps involved, so having this formula will help you solve perfect cube problems quickly.<br />
<br />
Let’s see some examples.<br />
<br />
First example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(x%2B1%5Cright)%5E%7B3%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(x+1)^3</span><br />
<br />
<span style="white-space: pre;"> </span> 1. Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=x</span>,<span class="mathquill-embedded-latex font16">b=1</span><br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)^3=x^3+3∙x^2∙1+3∙x∙1^2+1^3</span><br />
<span class="mathquill-embedded-latex font16">=x^3+3x^2+3x+1</span> <br />
<br />
Next example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(2x-2%5Cright)%5E%7B3%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(2x-2)^3</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=2x</span>,<span class="mathquill-embedded-latex font16">b=-2</span><br />
<br />
<span class="mathquill-embedded-latex font16">(2x-2)^3=(2x)^3+3∙(2x)^2∙(-2)+3∙(2x)∙(-2)^2+(-2)^3</span><br />
<span class="mathquill-embedded-latex font16">=8x^3-24x^2+24x-8</span> <br />
<br />
Last example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(s%5E%7B2%7D%2B2t%5Cright)%5E%7B3%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(s^2+2t)^3</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=s^2</span>,<span class="mathquill-embedded-latex font16">b=2t</span><br />
<br />
<span class="mathquill-embedded-latex font16">(s^2+2t)^3=(s^2 )^3+3∙(s^2 )^2∙(2t)+3∙s^2∙(2t)^2+(2t)^3</span><br />
<span class="mathquill-embedded-latex font16">=s^6+6s^4 t+12s^2 t^2+8t^3</span> <br />
<br />
Not too difficult! Memorizing this formula will help you and become faster at cubing binomials. If you are looking for more practice problems or help, check out Symbolab’s <span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/practice">Practice</a></span>.<br />
<br />
Until next time,<br />
<br />
LeahUnknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-77897624177778439082017-11-15T02:35:00.001-08:002022-06-08T07:57:01.678-07:00Middle School Math Solutions – Expand Calculator, Perfect SquaresThe perfect square formula is an application of the FOIL method that will help you calculate the square of a binomial quickly.<br />
<br />
Let’s take a look at the formula:<br />
<br />
<span class="mathquill-embedded-latex font16">(a+b)^2=a^2+2ab+b^2</span><br />
<br />
Here’s the proof:<br />
<br />
<span class="mathquill-embedded-latex font16">(a+b)^2=(a+b)(a+b)</span><br />
<span class="mathquill-embedded-latex font16">=a^2+ab+ab+b^2</span><br />
<span class="mathquill-embedded-latex font16">=a^2+2ab+b^2</span><br />
<br />
Now, let’s see some examples using the perfect squares formula.<br />
<br />
First example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(x%2B1%5Cright)%5E%7B2%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(x+1)^2</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=x</span>, <span class="mathquill-embedded-latex font16">b=1</span><br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)^2=x^2+2∙x∙1+1^2</span><br />
<span class="mathquill-embedded-latex font16">=x^2+2x+1</span><br />
<br />
Second example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(5-x%5Cright)%5E%7B2%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(5-x)^2</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=5</span>, <span class="mathquill-embedded-latex font16">b=-x</span><br />
<br />
<span class="mathquill-embedded-latex font16">(5-x)^2=5^2-2∙5∙(-x)+(-x)^2</span><br />
<span class="mathquill-embedded-latex font16">=25+10x+x^2</span><br />
<br />
Last example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(s%5E%7B2%7D%2B4p%5Cright)%5E%7B2%7D?or=blog">click
here</a></span>):<br />
<br />
Expand <span class="mathquill-embedded-latex font16">(s^2+4p)^2</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=s^2</span>, <span class="mathquill-embedded-latex font16">b=4p</span><br />
<br />
<span class="mathquill-embedded-latex font16">(s^2+4p)^2=(s^2)^2+2∙s^2∙4p+(4p)^2</span><br />
<span class="mathquill-embedded-latex font16">=s^4+8s^2 p+16p^2</span><br />
<br />
The more you practice these problems, the faster you will be able to do them. I’ve used this formula so much, that now, I don’t need it since I can do it all mentally. For more help or practice on this topic, check out Symbolab’s <span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/practice">Practice</a></span>.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-12455835458652143442017-11-07T03:36:00.001-08:002022-06-08T07:57:02.403-07:00Middle School Math Solutions – Expand Calculator, Two SquaresThe difference of two squares is an application of the FOIL method (refer to our blog post on the <a href="http://blog.symbolab.com/2017/10/middle-school-math-solutions-expand_11.html" target="_blank">FOIL method</a>). The difference of two squares is a number or term squared subtracted from another number or term squared. We get it when we multiply two binomials, where the terms in the binomials are the same, except one of the terms is subtracted instead of being added.<br />
<br />
Let’s see this in a formula:<br />
<span class="mathquill-embedded-latex font16">(a+b)(a-b)=a^2-b^2</span><br />
<br />
Here’s the proof using the FOIL method:<br />
<span class="mathquill-embedded-latex font16">(a+b)(a-b)=a^2-ab+ab-b^2</span><br />
<span class="mathquill-embedded-latex font16">=a^2-b^2</span><br />
<br />
Not too complicated, so let’s see some examples.<br />
<br />
First example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(x%2B2%5Cright)%5Cleft(x-2%5Cright)?or=blog">click
here</a></span>):<br />
Expand <span class="mathquill-embedded-latex font16">(x+2)(x-2)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the formula<br />
<span class="mathquill-embedded-latex font16">a=x</span>,<span class="mathquill-embedded-latex font16">b=2</span><br />
<br />
<span class="mathquill-embedded-latex font16">(x+2)(x-2)=x^2-2^2</span><br />
<span class="mathquill-embedded-latex font16">=x^2-4</span><br />
<br />
This next one is a little more complicated.<br />
<br />
Next example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(-y%2B2x%5Cright)%5Cleft(y%2B2x%5Cright)?or=blog">click
here</a></span>):<br />
Expand <span class="mathquill-embedded-latex font16">(-y+2x)(y+2x)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Rewrite the problem<br />
<span class="mathquill-embedded-latex font16">(-y+2x)(y+2x)=(2x-y)(2x+y)</span><br />
<span class="mathquill-embedded-latex font16">=(2x+y)(2x-y)</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Apply the formula<br />
<br />
<span class="mathquill-embedded-latex font16">a=2x</span>,<span class="mathquill-embedded-latex font16">b=y</span><br />
<br />
<span class="mathquill-embedded-latex font16">(2x+y)(2x-y)=(2x)^2-y^2</span><br />
<span class="mathquill-embedded-latex font16">=4x^2-y^2</span><br />
<br />
For our last example, we will see an application of the difference of two squares formula.<br />
<br />
Last example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(20%2B3%5Cright)%5Cleft(20-3%5Cright)?or=blog">click
here</a></span>):<br />
<span class="mathquill-embedded-latex font16">23∙17</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Rewrite the numbers<br />
<span class="mathquill-embedded-latex font16">(20+3)(20-3)</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Apply the formula<br />
<span class="mathquill-embedded-latex font16">a=20</span>,<span class="mathquill-embedded-latex font16">b=3</span><br />
<br />
<span class="mathquill-embedded-latex font16">(20+3)(20-3)=20^2-3^2</span><br />
<span class="mathquill-embedded-latex font16">=400-9</span><br />
<span class="mathquill-embedded-latex font16">=391</span><br />
<br />
As you can see, this formula is simple, but very helpful. If you need more help or practice with this formula, check out Symbolab’s <span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/practice">Practice</a></span>.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-73644417402650501022017-10-31T08:13:00.001-07:002022-06-08T07:57:03.097-07:00Middle School Math Solutions – Expand Calculator, Polynomial MultiplicationAs of right now, you should be familiar with the distributive law and the FOIL method. We will now use both of them and apply them to our next topic, polynomial multiplication.<br />
<br />
Recall: A polynomial is an expression of many (poly-) terms (-nomial) that are added and/or subtracted together. The terms include coefficients, variables, and POSITIVE exponents.<br />
<br />
Examples: <span class="mathquill-embedded-latex font16">4x^2</span> ,<span class="mathquill-embedded-latex font16">2x^3+5x^5</span> ,<span class="mathquill-embedded-latex font16">\frac{2}{3}x-1</span><br />
<br />
There are a variety of polynomial multiplication problems. They might look intimidating at first, but we will go over how to solve them step by step using the distributive law and FOIL method, so you can see how to do them easily.<br />
<br />
First example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%204x%5Cleft(3x%5E%7B2%7D%2B2x-1%5Cright)?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">4x(3x^2+2x-1)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Distribute the <span class="mathquill-embedded-latex font16">4x</span><br />
<br />
<span class="mathquill-embedded-latex font16">4x(3x^2+2x-1)=4x∙3x^2+4x∙2x+4x∙(-1)</span><br />
<blockquote class="tr_bq">
We applied the distributive law here for a polynomial with three terms <span class="mathquill-embedded-latex font16">(a(b+c+d)=ab+ac+ad)</span></blockquote>
<span style="white-space: pre;"> 2. </span>Simplify<br />
<br />
<span class="mathquill-embedded-latex font16">4x(3x^2+2x-1)=12x^3+8x^2-4x</span><br />
<blockquote class="tr_bq">
We multiplied the terms.</blockquote>
That one wasn’t bad. Let’s see another one.<br />
<br />
Next example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(2x%2B1%5Cright)%5Cleft(4x%2B3y%2B2%5Cright)?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">(2x+1)(4x+3y+2)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Distribute <span class="mathquill-embedded-latex font16">2x+1</span><br />
<br />
<span class="mathquill-embedded-latex font16">(2x+1)(4x+3y+2)=2x(4x+3y+2)+1(4x+3y+2)</span><br />
<blockquote class="tr_bq">
We distributed <span class="mathquill-embedded-latex font16">2x+1</span> to make the problem easier for us to solve.</blockquote>
<span style="white-space: pre;"> 2. </span>Use the distributive law<br />
<br />
<span class="mathquill-embedded-latex font16">2x(4x+3y+2)=2x∙4x+2x∙3y+2x∙2</span><br />
<br />
<span class="mathquill-embedded-latex font16">1(4x+3y+2)=4x∙1+3y∙1+2∙1</span><br />
<blockquote class="tr_bq">
We plug these values back in</blockquote>
<span class="mathquill-embedded-latex font16">(2x+1)(4x+3y+2)=2x∙4x+2x∙3y+2x∙2+4x∙1+3y∙1+2∙1</span><br />
<br />
<span style="white-space: pre;"> 3. </span>Simplify<br />
<br />
<span class="mathquill-embedded-latex font16">(2x+1)(4x+3y+2)=8x^2+6xy+4x+4x+3y+2</span><br />
<span class="mathquill-embedded-latex font16">=8x^2+6xy+8x+3y+2</span><br />
<br />
Last example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(x%2B1%5Cright)%5Cleft(x%2B2%5Cright)%5Cleft(x%2B3%5Cright)?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)(x+2)(x+3)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Apply the FOIL method to two of the polynomials<br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)(x+2)=x∙x+x∙2+1∙x+1∙2</span><br />
<span class="mathquill-embedded-latex font16">=x^2+3x+2</span><br />
<br />
<span style="white-space: pre;"> 2. </span>Plug <span class="mathquill-embedded-latex font16">x^2+3x+2</span> in for <span class="mathquill-embedded-latex font16">(x+1)(x+2)</span><br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)(x+2)(x+3)=(x^2+3x+2)(x+3)</span><br />
<blockquote class="tr_bq">
Now this problem looks like the prior example.</blockquote>
<span style="white-space: pre;"> 3. </span>Distribute x+3<br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)(x+2)(x+3)=x(x^2+3x+2)+3(x^2+3x+2)</span><br />
<blockquote class="tr_bq">
I always prefer to distribute the polynomial with the smallest terms, but you can pick to distribute the quadratic. In that case you would have:</blockquote>
<span class="mathquill-embedded-latex font16">(x^2+3x+2)(x+3)=x^2 (x+3)+3x(x+3)+2(x+3)</span><br />
<br />
<span style="white-space: pre;"> 4. </span>Use the distributive law<br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)(x+2)(x+3)=x(x^2+3x+2)+3(x^2+3x+2)</span><br />
<span class="mathquill-embedded-latex font16">=x∙x^2+x∙3x+x∙2+3∙x^2+3∙3x+3∙2</span><br />
<br />
<span style="white-space: pre;"> 5. </span>Simplify<br />
<br />
<span class="mathquill-embedded-latex font16">(x+1)(x+2)(x+3)=x^3+3x^2+2x+3x^2+9x+6</span><br />
<span class="mathquill-embedded-latex font16">=x^3+6x^2+11x+6</span><br />
<br />
Not bad! You can see when we break the problem down, step by step, it is a lot easier to solve because we use the basics of expanding (distributive law and FOIL). For more practice problems and help, check out our <span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/practice">Practice</a></span>.<br />
<br />
Until next time,<br />
<br />
Leah<br />
<div>
<br /></div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-3056318318610018494.post-91114252154759888642017-10-11T08:38:00.001-07:002022-06-08T07:57:04.170-07:00Middle School Math Solutions – Expand Calculator, FOIL MethodIn our <a href="http://blog.symbolab.com/2017/09/middle-school-math-solutions-expand.html" target="_blank">last blog post</a> we covered the distributive law. In this blog post, we will focus on an application of the distributive law, the FOIL method.<br />
<br />
The FOIL method is used when multiplying two binomials together. Quick reminder: a binomial is an expression of the sum or difference of two terms.<br />
<br />
The FOIL method is when we take the sum of the first two terms multiplied together, the outer terms multiplied together, the inner terms multiplied together, and the last terms multiplied together. This is where we get the acronym FOIL (first, outer, inner last).<br />
<br />
Let’s visualize this and see it in a formula:<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8GAp5CCTY4EqiWhR1olLeWkTMSEzVdprvZUb2ChWQgQ93n8t-gc6kiHrj-7nGBkGFQuyhiiT2ZWp66AJsE9whS6Q2ja2FyeJXBtX21ZBEhdrUlg9nElvm0cWUByi2TYgozAa_omduD7w/s1600/picsym30.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="144" data-original-width="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8GAp5CCTY4EqiWhR1olLeWkTMSEzVdprvZUb2ChWQgQ93n8t-gc6kiHrj-7nGBkGFQuyhiiT2ZWp66AJsE9whS6Q2ja2FyeJXBtX21ZBEhdrUlg9nElvm0cWUByi2TYgozAa_omduD7w/s1600/picsym30.png" /></a></div>
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Let’s see some examples now, using the FOIL method.<br />
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First example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(x%2B3%5Cright)%5Cleft(x%2B1%5Cright)?or=blog">click
here</a></span>):<br />
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<span class="mathquill-embedded-latex font16">(x+3)(x+1)</span><br />
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<span style="white-space: pre;"> </span>1. Use the FOIL method<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdqU45Qf45mFrHwVsQ1_wQJS0VpUAdX5V4O5pD9mvEdLNPTgKwCv8XX0L6uRwKBbu2DOK5pFPJmzn18mbzpilaeU7lGr19BJb85qNU9zfj088TjqymY2tOZAhDm-ta2_4tnz8VVpLmGXI/s1600/picsym31.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="144" data-original-width="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdqU45Qf45mFrHwVsQ1_wQJS0VpUAdX5V4O5pD9mvEdLNPTgKwCv8XX0L6uRwKBbu2DOK5pFPJmzn18mbzpilaeU7lGr19BJb85qNU9zfj088TjqymY2tOZAhDm-ta2_4tnz8VVpLmGXI/s1600/picsym31.png" /></a></div>
<blockquote class="tr_bq">
Remember, we want the sum of the first terms multiplied together, the outer terms multiplied together, the inner terms multiplied together, and the last terms multiplied together.</blockquote>
<span style="white-space: pre;"> </span>2. Simplify<br />
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<span class="mathquill-embedded-latex font16">(x+3)(x+1)=x∙x+x∙1+3∙x+3∙1</span><br />
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<span class="mathquill-embedded-latex font16">=x^2+x+3x+3</span><br />
<br />
<span class="mathquill-embedded-latex font16">=x^2+4x+3</span><br />
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<span style="white-space: pre;"> </span> Multiply terms and add like terms<br />
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Not, too complicated. Let’s see two more examples.<br />
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Next example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(2x%2B1%5Cright)%5Cleft(x-1%5Cright)?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">(2x+1)(x-1)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Use the FOIL method<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVTsiHwaAoZvSz9tQtrXV_dcSp3XC6B8zy4aekv32HzzMinbz1L76fzVYKvq_5z1E1HcHIC-Ew5CwPHe_BghaKfHfbcMqx6XJWaX5x_qrLsSs6w3sZavhxkIde1-2TeujFvbPEa8jJgtg/s1600/picsym32.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="143" data-original-width="358" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVTsiHwaAoZvSz9tQtrXV_dcSp3XC6B8zy4aekv32HzzMinbz1L76fzVYKvq_5z1E1HcHIC-Ew5CwPHe_BghaKfHfbcMqx6XJWaX5x_qrLsSs6w3sZavhxkIde1-2TeujFvbPEa8jJgtg/s1600/picsym32.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Simplify<br />
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<span class="mathquill-embedded-latex font16">(2x+1)(x-1)=2x^2-2x+x-1</span><br />
<br />
<span class="mathquill-embedded-latex font16">=2x^2-x-1</span><br />
<span class="mathquill-embedded-latex font16"><br /></span>
Last example (<span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/solver/step-by-step/expand%20%5Cleft(3x%5E%7B2%7D%2B2y%5Cright)%5Cleft(-x%2B5y%5Cright)?or=blog">click
here</a></span>):<br />
<br />
<span class="mathquill-embedded-latex font16">(3x^2+y)(-x+5y)</span><br />
<br />
<span style="white-space: pre;"> 1. </span>Use the FOIL method<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDlalWxWxlNEAq8sFaHgurgdrS5PiTtsG5CdNTAP9hQ-RRhitP_n1T8EfbXe74KkExV97RRYOaCU_S3ooPZShMv7FtRnxoqD2Wng3N5QDw-2dbeqV6t42dFYgZYZExc0PoV5AH8m8i9X0/s1600/picsym33.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="142" data-original-width="474" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDlalWxWxlNEAq8sFaHgurgdrS5PiTtsG5CdNTAP9hQ-RRhitP_n1T8EfbXe74KkExV97RRYOaCU_S3ooPZShMv7FtRnxoqD2Wng3N5QDw-2dbeqV6t42dFYgZYZExc0PoV5AH8m8i9X0/s1600/picsym33.png" /></a></div>
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<span style="white-space: pre;"> 2. </span>Simplify<br />
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<span class="mathquill-embedded-latex font16">(3x^2+y)(-x+5y)=-3x^3+15x^2 y-xy+5y^2</span><br />
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As you can see, multiplying binomials using the FOIL method isn’t hard. The FOIL method is so helpful throughout your math courses, so it is important to memorize and get the hang of it. Once you start practicing, this will become second nature to you. For practice problems or more help on the FOIL method, check out our <span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%;"><a href="https://www.symbolab.com/practice">Practice</a></span>.<br />
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Until next time,<br />
<br />
LeahUnknownnoreply@blogger.com