What is the inverse of a matrix?
The inverse of a matrix is like the reciprocal of a number. The inverse of an matrix is ,an matrix, such that , where is the identity matrix. Just like how the product of a number and its reciprocal equals 1, ( ), the product of a matrix and its inverse equals the identity matrix.
An identity matrix is an matrix, where the main diagonal of the matrix is all 1s and everywhere else in the matrix is 0s.
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 matrix with the identity matrix to create matrix
2. Put the matrix on the left hand side of the augmented matrix in RREF
 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
 The matrix on the left hand side will be converted to the identity matrix
3. The inverse matrix, , is to the right of the augmented matrix
 Whatever elementary row operations you do to the left matrix will be done to the matrix on the right
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
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.
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
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
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