Inverse of a Matrix

An n x n matrix A is said to have an inverse provided there exists an n x n matrix B such that AB = BA = I_n. We call B the inverse of A and denote it as A^-1. Thus, AA^-1 = A^-1A = I_n. In this case, A is also called nonsingular.

Example.

$ A = \left(\begin{array}{cccc}4&3\\3&2\end{array}\right) $ $ A^-1\left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right) $

Theorem 1

The inverse of a matrix, if it exists, is unique

Theorem 2

If A and B are both nonsingular n x n matrices (i.e. invertible), then AB is nonsingular and (AB^)-1 = B^-1A^-1.