# Inverse Matrix

The inverse of a matrix
**A** is said to be the matrix which when multiplied by **A** results in an
identity matrix. i.e.

where
denotes the inverse of **A**

An inverse matrix has the same size as the matrix of which it is an inverse. Not all matrices have inverses. When a matrix has an inverse, it is said to be invertible. A matrix is invertible if and only if its determinant is NOT zero. The reason for this will become clear when we see how the inverse of a matrix is obtained.

## Properties of Inverse Matrices

Given that matrix **A** is invertible, then **A** has the following properties:

- The determinant of
**A**is not zero - The determinant of the inverse of
**A**is the inverse of the determinant of**A** - The inverse of an Inverse of an inverse matrix is equal to the original matrix
- The inverse of a matrix that has been multiplied by a non-zero scalar (c) is equal
to the inverse of the scalar multiplied by the inverse of the matrix
- The inverse distributes evenly across matrix multiplication

## Inverse of a 2 x 2 Matrix

Given a matrix **A** of size 2 x 2 such that

The inverse of **A** can be found from the following formula:

which can also be written as

This is why a matrix with determinant zero can't have an inverse, you would end up dividing by zero!

Example: Find the inverse of **A** given that

*solution:*

The first step is to find the determinant of **A**

Next, find the inverse using the formula stated above