Row operations are given by multiplication by elementary matrices.

**Theorem.** Performing an elementary row operation on a matrix $A$ is equivalent to left-multiplying $A$ by the corresponding elementary matrix.

If $E$ is the elementary matrix obtained by applying a row operation to $I_n$, then applying that same row operation to any $n \times m$ matrix $A$ yields the product $EA$.

**Example.** To swap rows 1 and 2 of $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$, multiply by the elementary matrix $E_{12}$:
$$E_{12} A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} = \begin{pmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{pmatrix}$$

Similarly, column operations correspond to right-multiplication by elementary matrices.Copy