Determinant of Elementary Column Matrix/Scale Column and Add

Theorem
Let $e_2$ be the elementary column operation $\text {ECO} 2$:

which is to operate on some arbitrary matrix space.

Let $\mathbf E_2$ be the elementary column matrix corresponding to $e_2$.

The determinant of $\mathbf E_2$ is:
 * $\map \det {\mathbf E_2} = 1$

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

By definition of elementary column matrix, $\mathbf E_2$ is the result of applying $\text {ECO} 2$ to a unit matrix $\mathbf I$.

Thus its effect is to add a (scalar) multiple of one column of $\mathbf I$ to another column of $\mathbf I$.

By Multiple of Column Added to Column of Determinant, this means that:
 * $\map \det {\mathbf I \mathbf E_2} = \map \det {\mathbf I}$

Hence the result.