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.

Remarks

The method works because each row operation is left multiplication by an elementary matrix. If $E_k \cdots E_1 A = I_n$, then $E_k \cdots E_1 = A^{-1}$, and the same product changes $I_n$ into $A^{-1}$. For $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, the method gives $A^{-1} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$.

Explanations

L1 human

We write the identity matrix next to $A$ and row reduce the whole wide matrix. When the left half becomes the identity, the right half has become $A^{-1}$. The same row operations that turn $A$ into $I$ turn $I$ into $A^{-1}$.