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.Copy