Computing the inverse by row reduction

Summary

Row reduction computes the inverse of a square matrix, or shows that no inverse exists. Let $A$ be an $n \times n$ matrix over a field $F$, and form the $n \times 2n$ augmented matrix $[A \mid I_n]$. Apply Gauss–Jordan elimination to it. If the left block becomes $I_n$, then the result is $[I_n \mid A^{-1}]$, and the right block is the inverse. If the left block gets a zero row, then $A$ is not invertible.