Determinant with Rows Transposed/Proof 1

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


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:

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


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

Hence the result.