### Introduction

In number arithmetic every number $a$ ( $\ne 0$ ) has a reciprocal $b$ written as ${a}^{-1}$ or $\frac{1}{a}$ such that $ba=ab=1$ . Some, but not all, square matrices have inverses. If a square matrix $A$ has an inverse, ${A}^{-1}$ , then

$\phantom{\rule{2em}{0ex}}A{A}^{-1}={A}^{-1}A=I$ .

We develop a rule for finding the inverse of a $2\times 2$ matrix (where it exists) and we look at two methods of finding the inverse of a $3\times 3$ matrix (where it exists).

Non-square matrices do not possess inverses so this Section only refers to square matrices.

#### Prerequisites

- be familiar with the algebra of matrices
- be able to calculate a determinant
- know what a cofactor is

#### Learning Outcomes

- state the condition for the existence of an inverse matrix
- use the formula for finding the inverse of a $2\times 2$ matrix
- find the inverse of a $3\times 3$ matrix using row operations and using the determinant method

#### Contents

1 The inverse of a square matrix2 The inverse of a 2 $\times $ 2 matrix

3 The inverse of a 3 $\times $ 3 matrix - Gauss elimination method

4 The inverse of a 3 $\times $ 3 matrix - determinant method