Definition of row reduce a matrix

Definition: To row reduce a matrix means to apply a sequence of elementary row operations to transform it into row echelon form (or reduced row echelon form).

Three elementary row operations:

1. Row swap: Interchange two rows ($R_i \leftrightarrow R_j$).
2. Row scaling: Multiply a row by a nonzero scalar ($R_i \leftarrow cR_i$, $c \neq 0$).
3. Row replacement: Add a multiple of one row to another ($R_i \leftarrow R_i + cR_j$).

Two matrices are row-equivalent if one can be obtained from the other by row reduction. Row equivalence is an equivalence relation.