Matrix Inverse

We can define a special matrix that is a square matrix, an inverse of a matrix.

The inverse of an $n x n$ matrix A is an $nxn$ matrix B such that

$AB = BA = I$,

where I is the identity matrix. If the above equality holds, then B is the inverse of A ($\mathbf{A}^{-1}$

If matrix A has an invsrse, if is said to be a nonsingular matrix.

Computing the Inverse

It is possible to compute the inverse of a matrix using either elementary row operations (Gaussian Elimination really) or determinants.

Please see this wonderful video for a step by step computation. The mechanics of computation starts about 6 minutes in.

As a practice problem, please see this page. The linked videos (right hand column) illustrate the computation of the inverse using both elementary row operations and determinants.