Elementary Row Operations as Matrix Multiplications/Corollary

Theorem

Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary row operation.

Then there exists an elementary row matrix of order $m$ such that:

$\mathbf {E X} = \mathbf Y$

Proof

Let $e$ be the elementary row operation such that $e \paren {\mathbf X} = \mathbf Y$.

Then this result follows immediately from Elementary Row Operations as Matrix Multiplications:

$e \paren {\mathbf X} = \mathbf {E X} = \mathbf Y$

where $\mathbf E = e \paren {\mathbf I}$.

$\blacksquare$

Sources

• 1995: John B. Fraleigh and Raymond A. Beauregard: Linear Algebra (3rd ed.) $\S 1.4$