Matrix Inverse Algorithm

Algorithm
The purpose of this algorithm is to convert a matrix into its inverse, or to determine that such an inverse does not exist.

Let $\mathbf A$ be the $n \times n$ square matrix in question.

Let $\mathbf I$ be the identity matrix of order $n$.


 * Step 0: Create the augmented matrix $\left[{\mathbf A | \mathbf I}\right]$.


 * Step 1: Perform elementary row operations until $\left[{\mathbf A | \mathbf I}\right]$ is in reduced row echelon form.

Call this new augmented matrix $\left[{\mathbf H | \mathbf C}\right]$.


 * Step 2:


 * If $\mathbf H = \mathbf I$, then take $\mathbf C = \mathbf A^{-1}$.


 * If $\mathbf H \ne \mathbf I$, $\mathbf A$ is not invertible.

Proof
Follows from Transformation of Identity Matrix into Inverse and Matrix Row Equivalent to Reduced Echelon Matrix.