Gauss-Jordan elimination is an algorithm to solve a system of linear equation and find the inverse of an invertible matrix.

It first augments a square matrix with identity matrix, and reduces the augmented matrix to to reduced row echelon form:

\begin{aligned} &\begin{array}{cc|cc} 1 & 0 & 7 & -3 \ 0 & 1 & -2 & 1 \end{array} \ =\quad &\begin{array}{cc|cc} 1 & 3 & 1 & 0 \ 0 & 1 & -2 & 1 \end{array} \ =\quad &\begin{array}{cc|cc} 1 & 3 & 1 & 0 \ 2 & 7 & 0 & 1 \end{array} \end{aligned}

The algorithm can be seen as an extension to the Gaussian elimination.