Tuesday, April 24, 2018

Advanced Math Solutions - Matrix Multiply, Power Calculator, Matrix Multiplication

In a previous blog post, 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.

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.

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.

Let’s see what this looks like:


 



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 . . .

Now, let’s see some examples to help better understand how multiply matrices.

First example (click here):


1. Take the dot product of the rows of matrix A and the columns of matrix B


2. Simplify


Next example (click here):


1. Take the dot product of the rows of matrix A and the columns of matrix B


2. Simplify


Last example (click here):


1. Take the dot product of the rows of matrix A and the columns of matrix B


2. Simplify


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 Practice for practice problems and quizzes on this topic.

Until next time,

Leah

Saturday, March 31, 2018

High School Math Solutions - Matrix Transpose Calculator, Transpose

The 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 A^T.

When you transpose a matrix, the element in row i, column j becomes the element in row j, column i of the transposed matrix.

Let’s see some examples to better understand what the transpose of a matrix is.

First example (click here):


1. Turn the rows into columns


Next example (click here):


1. Turn the rows into columns


Last example (click here):


1. Turn the rows into columns


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 Practice.

Until next time,

Leah

Wednesday, March 21, 2018

High School Math Solutions - Matrix Multiply Calculator, Matrix Scalar Multiplication

Last 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.

Scalar multiplication is when you multiply a matrix by a value, called a scalar. In scalar multiplication, you multiply each element of the matrix by the scalar.

Here is what scalar multiplication looks like:


Pretty simple, right? Now, let’s see some examples.

First example (click here):


1. Multiply each of the matrix elements by the scalar


2. Simplify

Next example (click here):


1. Multiply each of the matrix elements by the scalar


2. Simplify

Last example (click here):


1. Multiply each of the matrix elements by the scalar


2. Simplify

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 Practice.

Until next time,

Leah

Sunday, March 11, 2018

High School Math Solutions - Matrix Add, Subtract Calculator, Matrices

A 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.

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.

Here is how to add matrices together:






You’d follow the same method for subtraction.

Here is an example of matrices that can’t be added together because they aren’t the same size:






You can see one matrix is 2x2 and the other is 2x3.

Let’s see some examples to better understand.

Here’s the first example (click here):


1. Add elements in the matching positions


2. Simplify


Next example (click here):


1. Add elements in the matching positions


2. Simplify


Last example (click here):


1. Add elements in the matching positions

2. Simplify


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 Practice.

Until next time,

Leah

Tuesday, January 16, 2018

Middle School Math Solutions – Order of Operations Calculator

Given a problem like, 3×5-2, you might be wondering “how do I solve this?” since there are two possible ways.

Option 1: (3×5)-2=15-2=13

Option 2: 3×(5-2)=3×3=9

This is where the order of operations comes in, since there is only one correct answer.

The order of operations are rules that tell us which operation to perform first when calculating an algebraic expression.

Here is the order of operations:

1.  Parentheses
2.  Exponents
3.  Multiplication and Division (from left to right)
4.  Addition and Subtraction (from left to right)

When calculating an algebraic expression, go down the order of operations, starting with parentheses, and perform the operations that apply.

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.

Looking back at the problem above and knowing the order of operations, we now can see that the answer is 13, option 1.

Let’s see some examples using the order of operations.

First example (click here):

                                                              8×5-(6+10)÷2

1. Calculate within parentheses

                                                               =8×5-16÷2
We simplified what was in the parentheses.
2. Multiplication and Division (from left to right)

                                                               =40-16÷2
                                                               =40-8
We multiplied and divided starting from the left of the expression, making our way to the right.
3. Addition and Subtraction (from left to right)

                                                                 =40-8
                                                                 =32

Next example (click here):

                                                       2×(5×6+2^2 )+(8÷2)

1. Calculate within parentheses – Evaluate (5×6+2^2)

                                                       (5×6+2^2 )=(5×6+4)
                                                                         =(30+4)
                                                                         =34
Make sure you apply the order of operations when calculating inside the parentheses
Now we have: 2×(5×6+2^2 )+(8÷2)=2×34+(8÷2)
2. Calculate within parentheses – Evaluate  (8÷2)

                                                                   8÷2=4
Now we have: 2×34+(8÷2)=2×34+4
3. Multiplication and Division (from left to right)

                                                                =2×34+4
                                                                =68+4

4. Addition and Subtraction (from left to right)

                                                                =68+4
                                                                =72

Last example (click here):

                                                        6+(2(4+2))^2-5^2

1. Calculate within parentheses

                                                        =6+(2(6) )^2-5^2
                                                        =6+12^2-5^2

2. Exponents

                                                           =6+144-5^2
                                                           =6+144-25

3. Addition and Subtraction (from left to right)

                                                            =150-25
                                                            =125

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 Practice.

Until next time,

Leah

Wednesday, January 3, 2018

Middle School Math Solutions – Expand Calculator, Binomial Expansion

We’ve learned how to expand perfect squares and perfect cubes. Now, we are going to learn how to expand binomials raised to any positive integers.

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!

Here is the Binomial Theorem:

                                            (a+b)^n=\sum_{i=0}^n\binom{n}{i}a^(n-i)b^i

\binom{n}{i} is a combination, which we read as “n choose i”.
Here is the formula for n choose i:

                                                        \binom{n}{i}=\frac{n!}{i!(n-i)!}

Let’s see some examples using this formula.

First example (click here):

                                                          Expand (x+2)^4

1. Apply the formula

                                                         a=x, b=2, n=4

                                            (x+2)^4=\sum_{i=0}^4\binom{4}{i}x^(4-i)2^i

2. Expand the summation and simplify

           \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

                                       =\frac{4!}{0!(4-0)!} x^4∙2^0+\frac{4!}{1!(4-1)!} x^3∙2^1 
                   
                                          +\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
           
                                 =\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

                                      =x^4+4x^3∙2+6x^2∙4+4x∙8+1∙16
                      
                                      =x^4+8x^3+24x^2+32x+16 
                                   
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).

Next example (click here):

                                                             Expand (x-y)^5

1. Apply the formula

                                                              a=x, b=-y, n=5

                                           (x-y)^5=\sum_{i=0}^5\binom{5}{i} x^(5-i) (-y)^i

2. Expand the summation and simplify

               \sum_{i=0}^5\binom{5}{i} x^(5-i) (-y)^i
                                  =\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
               =x^5-5x^4 y+10x^3 y^2-10x^2 y^3+5xy^4-y^5

Last example (click here):

                                                          Expand (3+x^2 )^4

1. Apply the formula

                                                          a=3, b=x^2, n=4

                                           (3+x^2)^4=\sum_{i=0}^4\binom{4}{i} 3^(4-i) (x^2)^i

2. Expand the summation and simplify

            \sum_{i=0}^4\binom{4}{i} 3^(4-i) (x^2 )^i
                                                                          =\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

                     =81+4∙27x^2+6∙9x^4+4∙3x^6+x^8
                                    
                     =81+108x^2+54x^4+12x^6+x^8
                                            
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 Practice.

Until next time,

Leah