elementary row operations

Swap, scale, or replace a row — the three legal, reversible moves on a matrix.

There are exactly three elementary row operations: swap two rows ( $R_i\leftrightarrow R_j$ ), scale a row by a nonzero constant ( $R_i\to cR_i$ ), and replace a row with itself plus a multiple of another ( $R_i\to R_i+cR_j$ ). Each is reversible, which is exactly why none of them can change a system's solution set. Two matrices are row equivalent if one is reachable from the other by a finite sequence of these.