Determinant of Elementary Column Matrix/Scale Column

Theorem
Let $e_1$ be the elementary column operation $\text {ECO} 1$:

which is to operate on some arbitrary matrix space.

Let $\mathbf E_1$ be the elementary column matrix corresponding to $e_1$.

The determinant of $\mathbf E_1$ is:
 * $\map \det {\mathbf E_1} = \lambda$

Proof
By Elementary Matrix corresponding to Elementary Column Operation: Scale Column, the elementary column matrix corresponding to $e_1$ is of the form:
 * $E_{a b} = \begin {cases} \delta_{a b} & : a \ne k \\ \lambda \cdot \delta_{a b} & : a = k \end{cases}$

where:
 * $E_{a b}$ denotes the element of $\mathbf E_1$ whose indices are $\tuple {a, b}$
 * $\delta_{a b}$ is the Kronecker delta:
 * $\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$

Thus when $a \ne b$, $E_{a b} = 0$.

This means that $\mathbf E_1$ is a diagonal matrix.