Multiple of Row Added to Row of Determinant/Proof 1

Proof
Let $e$ be the elementary row operation that adds $k$ times row $r$ to row $s$.

Let $\mathbf B = \map e {\mathbf A}$.

Let $\mathbf E$ be the elementary row matrix corresponding to $e$.

From Elementary Row Operations as Matrix Multiplications:
 * $\mathbf B = \mathbf E \mathbf A$

From Determinant of Elementary Row Matrix: Scale Row and Add:
 * $\map \det {\mathbf E} = 1$

Then:

Hence the result.