Advanced Math Solutions - Matrix Inverse Calculator, Matrices

In this blog post, I will talk about how to get the inverse of a matrix. Please refresh yourself on RREF before continuing, if you haven’t mastered this topic yet.

What is the inverse of a matrix?
The inverse of a matrix is like the reciprocal of a number. The inverse of an n\times n matrix A is A^{-1},an n\times n matrix, such that A⋅A^{-1}=I, where I is the identity matrix. Just like how the product of a number and its reciprocal equals 1, (\frac{1}{n}⋅n=1), the product of a matrix and its inverse equals the identity matrix.

An identity matrix is an n\times n matrix, where the main diagonal of the matrix is all 1s and everywhere else in the matrix is 0s.

Note: You can only calculate the inverse of a square (n\times n) matrix.

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.

How do you calculate the inverse of a matrix?
1. Augment the n\times n matrix A with the n\times n identity matrix to create matrix (A|I)

• Augmenting a matrix means to create a matrix by appending the columns of two matrices
• Augmenting the matrix allows you to perform the same elementary row operations on both sides of the matrix

2. Put the matrix on the left hand side of the augmented matrix in RREF

• The matrix on the left hand side will be converted to the identity matrix   

•   Whatever elementary row operations you do to the left matrix will be done to the matrix on the right

3. The inverse matrix, A^{-1}, is to the right of the augmented matrix

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!

First example

1. Augment the matrix with the identity matrix

2. Put the matrix on the left in RREF

                                                                        R_2-R_1→R_2

                                                                       \frac{1}{3} R_2→R_2

                                                                      R_1-R_2→R_1

                                                                       \frac{1}{2} R_1→R_1

3. The inverse matrix is on the right of the augmented matrix

You can verify that this is the inverse of the matrix by multiplying the inverse of the matrix and the matrix together (see here). If the product equals the identity matrix, then it is the inverse.

Next example (click here):

1. Augment the matrix with the identity matrix

2. Put the matrix on the left in RREF

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.

3. The inverse matrix is on the right of the augmented matrix

Last example (click here)

1. Augment the matrix with the identity matrix

2. Put the matrix on the left in RREF

3. The inverse matrix is on the right of the augmented matrix

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.

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.

For more help or practice on this topic, check out Symbolab’s Practice.
Next blog post, I’ll talk about what the determinant of a matrix is and how to calculate it.

Until next time,

Leah