Monday, March 25, 2019

Advanced Math Solutions - Matrix Rank Calculator, Matrices

In 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.

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.

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.

For example:


Here, Row 2 is a combination of Row 1 and Row 3 (Row 1 + Row 3). Therefore the rows are not linearly independent.

In order to determine the rank of a matrix:
1. Put the matrix in REF or RREF
2. Count the number of non-zero rows 
Let’s see some examples. Please note that I won’t be going over how to put the matrices in REF or RREF.

First example (click here):

1. Put the matrix in REF or RREF
                                                                The matrix is in RREF.
2. Count the number of non-zero rows
                                      There are 3 non-zero rows. The rank of this matrix is 3.

Not so bad! Next example (click here):


1. Put the matrix in REF or RREF
                                                               The matrix is in REF.
2. Count the number of non-zero rows
                                       There are 3 non-zero rows. The rank of this matrix is 3.

Last example (click here):

1. Put the matrix in REF or RREF
                                                              The matrix is in RREF.
2. Count the number of non-zero rows
                                     There are 4 non-zero rows. The rank of this matrix is 4.

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.

For more help or practice on this topic, check out Symbolab’s Practice.

Until next time,

Leah

Sunday, March 3, 2019

High School Math Solutions - Matrix Inverse Calculator, Matrices (Part 2)

In the last two blog posts, I talked about how to find the inverse of a matrix and how to calculate the determinant of the matrix. Please review these two blog posts before continuing, if you are not familiar with either topic.

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.

Here are the steps for calculating the inverse of a 2x2 matrix, using the shortcut:

1. Calculate the determinant of matrix A
                       Reminder:


2. Reorganize matrix A


3. Multiply matrix A by \frac{1}{det(A)}

These steps can be summarized by this formula:


Not too difficult, right? Let’s see some examples.

First example (click here):


1. Calculate the determinant of the matrix


2. Reorganize the matrix

3. Multiple the matrix by \frac{1}{det(A)}


Next example (click here):


1. Calculate the determinant of the matrix


2. Reorganize the matrix  

3. Multiple the matrix by \frac{1}{det(A)}


Last example (click here):


1. Calculate the determinant of the matrix


2. Reorganize the matrix  

3. Multiple the matrix by \frac{1}{det(A)}


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 Practice.

Until next time,

Leah

Tuesday, January 8, 2019

Advanced Math Solutions - Matrix Inverse Calculator, Determinants

Last blog 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.

The determinant is a value calculated from an n\times n 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).

We will discuss three ways to solve three different types of matrices.

Determinant of a 2x2 matrix:

Determinant of a 3x3 matrix:


Determinant of a nxn matrix that is 4x4 or larger:

1. Put the matrix in REF (here is the blog post on REF for reference)
  • Make note of how many times you swapped rows to achieve putting the matrix in REF 
2. Calculate the product of the elements in the diagonal
  • If no rows were swapped, this is the determinant
3. 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.
  • Swapping rows changes the sign of the determinant

Let’s see some examples to better understand how to calculate the determinant.

First example (click here):


1. Use this formula:



Next example (click here):


1. Use this formula: 


Last example (click here):


1. Put the matrix in REF


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.

Note that rows were swapped 3 times to achieve putting the matrix in REF.

2. Calculate the product of the diagonal


3. If rows were swapped, multiply the product of the diagonal by (-1) raised to the number of times rows were swapped.


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 Practice. Next blog post, I will talk about a shortcut for calculating the inverse of a 2x2 matrix, using its determinant.
Until next time,

Leah