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
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 $1.13$