Determinant with Rows Transposed/Proof 1

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

Let $1 \le r < s \le n$.

Let $e$ be the elementary row operation that exchanging rows $r$ and $s$.

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: Exchange Rows:
 * $\map \det {\mathbf E} = -1$

Then:

Hence the result.