REF, RREF, and the Three Row Operations
Swap two rows
Rᵢ ↔ RⱼMove a usable pivot into the next pivot position.
Scale a row
Rᵢ ← cRᵢ, c ≠ 0This calculator scales each pivot to 1 for a clearer echelon form.
Add a row multiple
Rᵢ ← Rᵢ + cRⱼEliminate entries below a pivot for REF and both below and above it for RREF.
10⁻⁴⁰ through 10⁴⁰ in magnitude. The selected relative tolerance controls numerical pivot decisions; use exact-arithmetic software when exact rational output is required.How to Use the Row Reduction Calculator
- Select the row and column counts, then enter every value or paste a complete rectangular matrix.
- Mark the final column as augmented when the matrix represents a linear system
[A | b]. - Choose REF for forward Gaussian elimination or RREF for full Gauss–Jordan elimination.
- Select Reduce matrix. Review the rank, pivot and free columns, system classification, and recorded steps.
Worked 3×4 Augmented-Matrix Example
The default matrix represents three equations in three unknowns:
Starting matrix [A | b]
[[1, 2, −1 | −4], [2, 3, −1 | −11], [−2, 0, −3 | 22]]RREF
[[1, 0, 0 | −8], [0, 1, 0 | 1], [0, 0, 1 | −2]]The three coefficient columns are pivots, so the coefficient rank equals the number of variables. The system has the unique solution x = −8, y = 1, and z = −2.
Reading Rank, Pivots, and System Status
A pivot column contains a leading entry in an echelon form. The number of pivots is the matrix rank. Non-pivot coefficient columns are free columns; in a consistent system, each free variable contributes a parameter to the solution family.
For an augmented matrix [A | b], the system is inconsistent when reduction produces a row of the form [0 0 … 0 | c] with nonzero c. Otherwise, it has a unique solution when every variable column is a pivot and infinitely many solutions when at least one variable column is free.
Row Reduction Calculator FAQs
What is the difference between REF and RREF?
REF has zeros below each pivot and places successive pivots farther right. RREF additionally makes each pivot 1 and clears every other entry in its pivot column.
Is reduced row echelon form unique?
Yes. Every matrix has one unique RREF. REF is not unique, so another valid elimination order may give a different echelon matrix with the same rank and pivot structure.
Can I row-reduce a non-square matrix?
Yes. This calculator accepts rectangular matrices from 1×1 through 8×9. A matrix does not need to be square to have REF, RREF, rank, pivots, or free columns.
How does the augmented-column option work?
The last column is treated as the constants vector rather than a variable column. It still participates in every row operation, but the calculator does not choose a coefficient pivot from it.
Does row reduction preserve a linear system's solutions?
Yes. Each elementary row operation is reversible, so it replaces the system with an equivalent system having the same solutions.
Why does a decimal result sometimes show a tiny error?
Most decimal fractions are approximated in binary floating point. The selected tolerance suppresses numerically negligible pivot candidates, but poorly scaled or sensitive matrices may still require more specialized numerical software.
Are my matrix entries private?
Yes. The page parses and reduces the matrix locally, and creates copied text and downloads in your browser. The calculator code does not transmit or save entered values.
Calculation Notes
Last reviewed: August 5, 2026 by the Starlight Tools editorial team.
The algorithm selects the strongest available pivot relative to each candidate row, scales pivots to 1, and records every effective elementary row operation. REF output depends on the pivot strategy; RREF is mathematically unique, subject to the selected numerical tolerance and floating-point rounding.