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 $\mathbf E = e \left({\mathbf I}\right)$ where $\mathbf I$ is the unit matrix.

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


 * $e \left({\mathbf A}\right) = \mathbf{E A}$

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

Proof
Let $s,t \in [1,2,\cdots,m]$ such that $s \ne t$.

Case I
For when $e$ is the elementary row operation $r_s \to cr_s$:
 * $E_{ik} =

\begin{cases} \delta_{ik} & :i \ne s \\ c \delta_{ik} & :i = s \end{cases}$ where $\delta$ is defined as the kronecker delta


 * $\displaystyle \left({\mathbf {EA}}\right)_{ij} = \sum_{k \mathop = 1}^m \mathbf E_{ik} \mathbf A_{kj} =

\begin{cases} \mathbf A_{ij} & :i \ne r \\ \alpha \mathbf A_{ij} & :i = r \end{cases}$

Case II
For when $e$ is the elementary row operation $r_s \to r_s + cr_t$:
 * $E_{ik} =

\begin{cases} \delta_{ik} & :i \ne s \\ \delta_{sk} + c \delta_{tk} & :i = s \end{cases}$ where $\delta$ is defined as the kronecker delta


 * $\displaystyle \left({\mathbf {EA}}\right)_{ij} = \sum_{k \mathop = 1}^m \mathbf E_{ik} \mathbf A_{kj} =

\begin{cases} \mathbf A_{ij} & :i \ne s \\ \mathbf A_{ij} + c \mathbf A_{tj} & :i = s \end{cases}$

Case III
For when $e$ is the elementary row operation $r_s \leftrightarrow r_t$:

By Elementary Row Operation as Sequence of Other Two:
 * $e_1e_2e_3e_4\left({\mathbf A}\right) = \mathbf E_1 \mathbf E_2 \mathbf E_3 \mathbf E_4 \mathbf A = \mathbf E \left({\mathbf A}\right) = e \left({\mathbf A}\right)$