Elementary Matrix corresponding to Elementary Row Operation

Theorem
Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$.

Let $e$ be an elementary row operation on $\mathbf I$.

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

where $\mathbf I$ is the unit matrix.

Let $r_k$ denote the $k$th row of $\mathbf I$ for $1 \le k \le m$.

Case $(3)$: Exchange Rows
Throughout the above:
 * $E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$
 * $\delta_{a b}$ is the Kronecker delta:
 * $\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$

Also see

 * Elementary Matrix corresponding to Elementary Column Operation