# Determinant of Elementary Column Matrix

## Theorem

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

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

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

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

 $(\text {ECO} 1)$ $:$ $\ds \kappa_k \to \lambda \kappa_k$ For some $\lambda \ne 0$, multiply column $k$ by $\lambda$

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:

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

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

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

 $(\text {ECO} 2)$ $:$ $\ds \kappa_i \to \kappa_i + \lambda \kappa_j$ For some $\lambda$, add $\lambda$ times column $j$ to column $i$

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:

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

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

Let $e_3$ be the elementary column operation $\text {ECO} 3$:

 $(\text {ECO} 3)$ $:$ $\ds \kappa_i \leftrightarrow \kappa_j$ Exchange columns $i$ and $j$

which is to operate on some arbitrary matrix space.

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

The determinant of $\mathbf E_3$ is:

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