Elementary Column Operations as Matrix Multiplications

Theorem
Let $e$ be an elementary column operation.

Let $\mathbf E$ be the elementary column matrix of order $n$ defined as:
 * $\mathbf E = e \paren {\mathbf I}$

where $\mathbf I$ is the unit matrix.

Then for every $m \times n$ matrix $\mathbf A$:


 * $e \paren {\mathbf A} = \mathbf A \mathbf E$

where $\mathbf A \mathbf E$ denotes the conventional matrix product.

Proof
Let $s, t \in \closedint 1 m$ such that $s \ne t$.

Case $1$
Let $e$ be the elementary column operation $\kappa_s \to \lambda \kappa_s$:
 * $E_{k i} = \begin{cases}

\delta_{k i} & : i \ne s \\ \lambda \delta_{k i} & : i = s \end{cases}$ where $\delta$ denotes the Kronecker delta.

Then:

Case $2$
Let $e$ be the elementary column operation $\kappa_s \to \kappa_s + \lambda \kappa_t$:
 * $E_{k i} = \begin {cases}

\delta_{k i} & : i \ne s \\ \delta_{k s} + \lambda \delta_{k t} & : i = s \end {cases}$ where $\delta$ denotes the Kronecker delta.

Then:

Case $3$
Let $e$ be the elementary column operation $\kappa_s \leftrightarrow \kappa_t$:

By Exchange of Columns as Sequence of Other Elementary Column Operations, this elementary column operation can be expressed as:
 * $\paren {e_1 e_2 e_3 e_4 \mathbf A} = e \paren {\mathbf A}$

where the $e_i$ are elementary column operation of the other two types.

For each $e_i$, let $\mathbf E_i = e_i \paren {\mathbf I}$.

Then:

Also see

 * Elementary Row Operations as Matrix Multiplications