RREF Calculator (Reduced Row Echelon Form)
Convert any matrix to Reduced Row Echelon Form using Gauss-Jordan elimination. Works for rectangular and square matrices of any size up to 6×6.
Notes
What is RREF?
A matrix is in Reduced Row Echelon Form (RREF) if: (1) all zero rows are at the bottom, (2) the leading entry (pivot) of each non-zero row is 1, (3) each pivot is to the right of the pivot above it, and (4) all other entries in the pivot column are 0.
Step-by-Step Example
Starting matrix (used as augmented system [A|b]):
| 2 | 1 | 5 |
| 4 | 3 | 11 |
R2 → R2 − 2·R1:
| 2 | 1 | 5 |
| 0 | 1 | 1 |
R1 → R1/2, then R1 → R1 − R2:
| 1 | 0 | 2 |
| 0 | 1 | 1 |
RREF complete. Solution: x=2, y=1.
- Matrix Calculations Guide — RREF — In-depth notes on RREF with worked examples
Frequently Asked Questions
What is the difference between REF and RREF?
Row Echelon Form (REF) has leading entries that increase left-to-right, but above-pivot entries may be nonzero. RREF additionally requires above-pivot entries to be zero and each pivot to equal 1.
Can I use RREF to solve Ax = b?
Yes. Augment A with b as the last column, compute RREF, and read off the solution. The calculator works on the full augmented matrix.
How does RREF reveal the rank?
The rank equals the number of pivot rows (rows with a leading 1) in the RREF. Zero rows at the bottom indicate the matrix is rank-deficient.