For normal multiplication, there exists an inverse equation for every number equation (except zero), such that equation. This of course is equation, since equation. Similar to this, given a square matrix equation, there may be an inverse matrix equation such that equation where equation is the identity matrix of the same order (same number of rows and columns as equation). 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 equation and its inverse equation are multiplied does not matter:

    equation

Given two invertible matrices equation and equation, the inverse of their product is equal to the product of the inverses:

    equation

The order in which a matrix is inverted and transposed does not matter:

    equation

Finding the inverse

We explain the procedure using the following sample matrix for which we want to find the inverse:

    equation

We start by writing down our matrix next to an identity matrix, like so:

    equation

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

    equation

Row2 = Row2 + 2 * Row1
Row3 = Row3 * (-1)

    equation

We swap row 2 and row 3 and multiply row 2 by -1 in the process:
Row2 = Row3
Row3 = Row2 * (-1)

    equation

Row1 = Row1 + 2 * Row3
Row2 = Row2 – Row3

    equation

We found our inverse matrix on the right side, which is:

    equation

Indeed, we can confirm this is correct, by multiplying our original matrix with the inverse matrix to get the identity matrix:

    equation