Determinant of Elementary Row Matrix/Scale Row and Add

Theorem
Let $e_2$ be the elementary row operation $\text {ERO} 2$:

which is to operate on some arbitrary matrix space.

Let $\mathbf E_2$ be the elementary row 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 row matrix, $\mathbf E_2$ is the result of applying $\text {ERO} 2$ to a unit matrix $\mathbf I$.

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

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

Hence the result.