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).
Nonsquare 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