Elementary Row Operations as Matrix Multiplications/Corollary
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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.) $\S 1.4$