# Determinant with Rows Transposed/Proof 1

## Theorem

If two rows of a matrix with determinant $D$ are transposed, its determinant becomes $-D$.

## 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 $e$.

$\mathbf B = \mathbf E \mathbf A$
$\map \det {\mathbf E} = -1$

Then:

 $\ds \map \det {\mathbf B}$ $=$ $\ds \map \det {\mathbf E \mathbf A}$ Determinant of Matrix Product $\ds$ $=$ $\ds -\map \det {\mathbf A}$ as $\map \det {\mathbf E} = -1$

Hence the result.

$\blacksquare$