Determinant of Elementary Row Matrix

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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:

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


Also see


Sources