For normal multiplication, there exists an inverse for every number (except zero), such that . This of course is , since . Similar to this, given a square matrix , there may be an inverse matrix such that where is the identity matrix of the same order (same number of rows and columns as ). Not every matrix has an inverse and those matrices that do have one are called invertible.

First we show you some rules and then we explain how you can find the inverse for a matrix.

Rules

The order in which the matrix and its inverse are multiplied does not matter: Given two invertible matrices and , the inverse of their product is equal to the product of the inverses: The order in which a matrix is inverted and transposed does not matter: Finding the inverse

We explain the procedure using the following sample matrix for which we want to find the inverse: We start by writing down our matrix next to an identity matrix, like so: We now do row operations to transform the left side to the identity matrix and apply the same operations to the right side. Once we successfully turned the left side into the identity matrix, the right side will be the inverse matrix that we are looking for. There are different ways you can solve this and we show how we solved it step by step:

Row3 = Row3 – Row1 Row2 = Row2 + 2 * Row1
Row3 = Row3 * (-1) We swap row 2 and row 3 and multiply row 2 by -1 in the process:
Row2 = Row3
Row3 = Row2 * (-1) Row1 = Row1 + 2 * Row3
Row2 = Row2 – Row3 We found our inverse matrix on the right side, which is: Indeed, we can confirm this is correct, by multiplying our original matrix with the inverse matrix to get the identity matrix: 