RREF calculator — reduced row echelon form
Row-reduce any matrix to reduced row echelon form and see every elementary row operation, written as R2 → R2 − 3·R1.
Enter your matrix, open the Decompositions tab and press RREF(A). The steps panel opens automatically and lists every operation in standard notation, so you can copy the working straight into your homework and check where your own attempt diverged.
What reduced row echelon form is
A matrix is in RREF when every leading entry is 1, each pivot is the only non-zero number in its column, and the pivots move strictly right as you go down. The crucial property is uniqueness: a given matrix has exactly one RREF, no matter which order you performed the row operations in. That makes it a reliable answer to compare against.
Worked example
| 1 | 2 | 1 |
| 2 | 4 | 3 |
| 3 | 6 | 5 |
| 1 | 2 | 0 |
| 0 | 0 | 1 |
| 0 | 0 | 0 |
The second column never gets a pivot, so its variable is free — and the zero row confirms the third equation carried no new information. Reading the form back: the rank is 2, and a system with this coefficient matrix has infinitely many solutions with one free parameter.
Common mistakes
- Stopping at row echelon form. Zeroing only below each pivot gives REF. RREF also clears everything above.
- Leaving a pivot different from 1. The reduced form requires dividing each pivot row by its pivot.
- Reordering columns. Swapping rows is allowed; swapping columns changes which variable is which.
- Missing the inconsistent row. A row reading
0 0 0 | cwith c ≠ 0 means the system has no solution at all.