Title: 7_4 The Inverse of a Matrix
Course: HELM Workbooks 2023-2024 > Section: HELM 7: Matrices

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

 \(\qquad AA^{-1}=A^{-1}A=I\).

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.