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 $r_i = \tuple {\mathbf A_{i, 1}, \mathbf A_{i, 2}, \cdots, \mathbf A_{i, n} }$, and let $r_j$ be defined likewise.

Then the elementary row operation $r_i \leftrightarrow r_j$ can be written as a finite sequence of exactly four of the other two types of elementary row operations.

Proof
We start with $\tuple {r_i, r_j}$


 * $r_j \to r_j + r_i: \quad \tuple {r_i, r_i + r_j}$


 * $r_i \to r_i + \tuple {-r_j}: \quad \tuple {-r_j, r_i + r_j}$


 * $r_j \to r_j + r_i: \quad \tuple {-r_j, r_i}$

And finally,


 * $r_i \to -r_i: \quad \tuple {r_j, r_i}$