Elementary Row Operations as Matrix Multiplications

Theorem
Let $e$ be an elementary row operation.

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

where $\mathbf I$ is the unit matrix.

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


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

where $\mathbf {E A}$ 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 row operation $r_s \to \lambda r_s$:
 * $E_{ik} = \begin{cases}

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

Then:

Case $2$
Let $e$ be the elementary row operation $r_s \to r_s + \lambda r_t$:
 * $E_{i k} = \begin{cases}

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

Then:

Case $3$
Let $e$ be the elementary row operation $r_s \leftrightarrow r_t$:

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

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

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

Then: