# 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, *A**A*^{-1} = *A*^{-1}*A* = *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) $

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^{-1}A = </math>$ \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*^{-1}*A*^{-1}.