Determinant of Elementary Column Matrix/Exchange Columns

Theorem
Let $e_3$ be the elementary column operation $\text {ECO} 3$:

which is to operate on some arbitrary matrix space.

Let $\mathbf E_3$ be the elementary column matrix corresponding to $e_3$.

The determinant of $\mathbf E_3$ is:
 * $\map \det {\mathbf E_3} = -1$

Proof
By Determinant of Unit Matrix:
 * $\map \det {\mathbf I} = 1$

By definition of elementary column matrix, $\mathbf E_3$ is the result of applying $\text {ECO} 3$ to a unit matrix $\mathbf I$.

Thus its effect is to exchange the positions of two of the columns of $\mathbf I$.

By Determinant with Columns Transposed, this means that:
 * $\map \det {\mathbf E_3 \mathbf I} = -\map \det {\mathbf I}$

Hence the result.