Exchange of Rows as Sequence of Other Elementary Row Operations

Theorem
Let $\mathbf A$ be an $m \times n$ matrix.

Let $i, j \in \closedint 1 m: i \ne j$

Let $r_k$ denote the $k$th row of $\mathbf A$ for $1 \le k \le m$:
 * $r_k = \begin {pmatrix} a_{k 1} & a_{k 2} & \cdots & a_{k n} \end {pmatrix}$

Let $e$ be the elementary row operation acting on $\mathbf A$ as:

Then $e$ can be expressed as a finite sequence of exactly $4$ instances of the other two elementary row operations.

Proof
In the below:
 * $r_i$ denotes the initial state of row $i$
 * $r_j$ denotes the initial state of row $j$


 * $r_i'$ denotes the state of row $i$ after having had the latest elementary row operation applied
 * $r_j'$ denotes the state of row $j$ after having had the latest elementary row operation applied.

$(1)$: Apply $\text {ERO} 2$ to row $j$ for $\lambda = 1$:


 * $r_j \to r_j + r_i$

After this operation:

$(2)$: Apply $\text {ERO} 2$ to row $i$ for $\lambda = -1$:


 * $r_i \to r_i + \paren {-r_j}$

After this operation:

$(3)$: Apply $\text {ERO} 2$ to row $j$ for $\lambda = 1$:


 * $r_j \to r_j + r_i$

After this operation:

$(4)$: Apply $\text {ERO} 1$ to row $i$ for $\lambda = -1$:


 * $r_i \to -r_i$

After this operation:

Thus, after all the $4$ elementary row operations have been applied, we have:

Hence the result.