# Determinant of Elementary Row Matrix

## Theorem

Let $\mathbf E$ be an elementary row matrix.

The determinant of $\mathbf E$ is as follows:

### Case $(1)$: Scalar Product of Row

Let $e_1$ be the elementary row operation $\text {ERO} 1$:

 $(\text {ERO} 1)$ $:$ $\displaystyle r_k \to \lambda r_k$ For some $\lambda \ne 0$, multiply row $k$ by $\lambda$

which is to operate on some arbitrary matrix space.

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

The determinant of $\mathbf E_1$ is:

$\map \det {\mathbf E_1} = \lambda$

### Case $(2)$: Add Scalar Product of Row to Another

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

 $(\text {ERO} 2)$ $:$ $\displaystyle r_i \to r_i + \lambda r_j$ For some $\lambda$, add $\lambda$ times row $j$ to row $i$

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:

$\map \det {\mathbf E_2} = 1$

### Case $(3)$: Exchange Rows

Let $e_3$ be the elementary row operation $\text {ERO} 3$:

 $(\text {ERO} 3)$ $:$ $\displaystyle r_i \leftrightarrow r_j$ Exchange rows $i$ and $j$

which is to operate on some arbitrary matrix space.

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

The determinant of $\mathbf E_3$ is:

$\map \det {\mathbf E_3} = -1$