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:

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

Also see

• Determinant of Elementary Row Matrix

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace: Proposition $1.13$