Leading entries become 1, pivot columns are cleared, and zero rows move to the bottom. Work across the entire row each time.
| 1 | 2 | 3 |
| 2 | 5 | 8 |
Make space for the next pivot.
The first leading entry is already 1. Subtract twice row 1 from row 2 to clear the entry below the first pivot.
| 1 | 2 | 3 |
| 0 | 1 | 2 |
Clear above the second pivot.
The second pivot is now 1. Subtract twice row 2 from row 1.
| 1 | 0 | −1 |
| 0 | 1 | 2 |
Check the result.
Both pivot columns contain a single 1 and zeros elsewhere. The matrix has rank 2, and its third column is a non-pivot column.
What makes a matrix RREF?
- Any rows made entirely of zeros are at the bottom.
- The first nonzero entry in each nonzero row, its pivot, is 1.
- Each pivot sits to the right of the pivot in the row above.
- Every other entry in a pivot column is 0.
A matrix that meets only the first and third conditions is in row echelon form. Compare the two in REF vs. RREF.
Know the three row operations.
- Swap two rows when a useful pivot sits below the current row.
- Multiply a row by a nonzero number to make its pivot 1.
- Add a multiple of one row to another to clear a pivot column.
Apply each operation across the entire row. For a system of equations, that includes the constants column. Keep fractions exact until the end.
Try a different matrix.
Open the RREF calculator to see every intermediate matrix for your own values. If you are solving equations, use the augmented matrix calculator and put the constants in the last column.
Reference: University of Texas at Austin, Matrix Operations appendix. Examples independently calculated. Source checked September 24, 2026.