Determinant with Row Multiplied by Constant/Proof 1

Proof
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $e$ be the elementary row operation that multiplies rows $i$ by the scalar$c$.

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: Exchange Rows:
 * $\map \det {\mathbf E} = c$

Then:

Hence the result.