Elementary Row Operations as Matrix Multiplications

Theorem
Let $\mathbf{X}$ be and $\mathbf{Y}$ be matrices that differ by exactly one elementary row operation.

Then and only then there exists some elementary matrix such that:


 * $\mathbf{EX} = \mathbf{Y}$

$\mathbf{E}$ is the identity matrix transformed by said elementary row operation.