# 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 = In. We call B the inverse of A and denote it as A-1. Thus, AA-1 = A-1A = In. In this case, A is also called nonsingular.

Example.

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

AA-1 = $\left(\begin{array}{cccc}4&3\\3&2\end{array}\right) $$\left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right) = \left(\begin{array}{cccc}1&0\\0&1\end{array}\right) and A-1A = [/itex] \left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right)$$ \left(\begin{array}{cccc}4&3\\3&2\end{array}\right)$ = $\left(\begin{array}{cccc}1&0\\0&1\end{array}\right)$

### 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.

## Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva