| Property | REF | RREF |
|---|---|---|
| Zero rows are last | Yes | Yes |
| Leading entries move right | Yes | Yes |
| Leading entries are 1 | Not required | Required |
| Zeros above pivots | Not required | Required |
| Unique result | No | Yes |
The difference you can see.
| 1 | 2 | 3 |
| 0 | 1 | 2 |
| 1 | 0 | −1 |
| 0 | 1 | 2 |
This calculator normalizes REF pivots to 1 as a convenient convention. REF does not require that normalization under the definition used here.
Row echelon form vs. reduced row echelon form.
Row echelon form (REF)
Zero rows sit at the bottom, and each leading entry is to the right of the one above it, with zeros below every leading entry. Many different REF matrices can come from the same starting matrix, depending on the row operations you choose.
Reduced row echelon form (RREF)
RREF adds two conditions: every leading entry is 1, and it is the only nonzero entry in its column. Those extra conditions make the reduced form unique for each matrix.
Which form should you use?
REF is enough for back-substitution when you solve equations by hand. RREF makes pivot and free-variable relationships direct to read, which is why it is the usual final answer. Both preserve the solution set when applied to an augmented system.
Compare them on your own example.
Use the Form menu in the RREF calculator. Changing the selection clears the previous result so you can calculate the new form explicitly and compare the row operations.
Reference: University of Texas at Austin, Matrix Operations appendix. Examples independently calculated. Source checked September 24, 2026.