Math
RREF Calculator
Type a matrix to get its reduced row echelon form (RREF) by Gauss-Jordan elimination, with the row echelon form, pivot columns, rank, and every row operation recorded.
Switch to Solve Ax = b mode to enter a coefficient matrix and constants: the calculator reports a unique solution, a parametric family with free variables, or explains why there is no solution.
One row per line. To reduce an augmented matrix, include the constants as the last column, or use Solve Ax = b mode.
Try an example
Result
RREF / solution
—
- Rank
- —
- Pivot columns
- —
- Free columns
- —
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What reduced row echelon form is
A A rectangular grid of numbers arranged in rows and columns. is in reduced row echelon form when every nonzero row starts with a leading 1 (a pivot), each pivot sits to the right of the pivot above it, every pivot is the only nonzero entry in its column, and all-zero rows are at the bottom. Row echelon form (REF) drops the requirements that pivots be 1 and that entries above pivots be zero. Every matrix has exactly one RREF.
The three row operations
| Operation | Notation | Effect on a linear system |
|---|---|---|
| Swap two rows | R1 <-> R2 | Reorders the equations |
| Multiply a row by a nonzero constant | R2 -> (1/3)·R2 | Scales one equation |
| Add a multiple of one row to another | R3 -> R3 - 2·R1 | Eliminates a variable from one equation |
How to use it
- Choose Row-reduce a matrix to see REF and RREF of any matrix, or Solve Ax = b to work with a The number multiplying a variable, like the 3 in 3x. matrix and constants.
- Enter the matrix one row per line. In Solve mode, put one constant per equation in the second box.
- Read the pivot columns and rank in the panel, then open Row operations to follow every step.
- In Solve mode, read the solution A quantity with both a size and a direction, often drawn as an arrow., the parametric form with free variables, or the reason no solution exists.
How to read the answer for Ax = b
| What the RREF shows | Meaning | Solutions |
|---|---|---|
| Pivot in every variable column, no contradictory row | rank = number of unknowns | Exactly one |
| Some variable column has no pivot | Free variable(s) | Infinitely many; written with parameters t |
| A row [0 0 … 0 | c] with c ≠ 0 | 0 = c, impossible | None (inconsistent) |
Common mistakes
- Stopping at row echelon form when the question asks for reduced row echelon form. RREF also needs zeros above every pivot.
- Multiplying a row by 0. That destroys an equation and is not an allowed operation.
- Reading a free variable as 'no solution'. A free variable means infinitely many solutions, unless a contradictory row also appears.
- Forgetting to apply each row operation to the constants column in an augmented matrix.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
Augmented 3 x 4 matrix
Rows encode x + 2y - z = -4, 2x + 3y - z = -11, -2x - 3z = 22
RREF / solution
[1, 0, 0, -8; 0, 1, 0, 1; 0, 0, 1, -2]
Solve Ax = b, unique solution
Same system in Solve mode: x = -8, y = 1, z = -2
RREF / solution
(-8, 1, -2)
Infinitely many solutions
x + y + z = 3 and 2x + 2y + 2z = 6 are the same plane
RREF / solution
Infinitely many: x = 3 - t1 - t2, y = t1, z = t2
No solution
x + y = 1 and x + y = 2 contradict
RREF / solution
No solution
Rank-deficient matrix
Row 3 = row 1 + row 2, so rank is 2
RREF / solution
[1, 0, -1; 0, 1, 2; 0, 0, 0]
Fractions
A matrix with 1/2 and 1/3 reduces exactly
RREF / solution
[1, 0; 0, 1]
Zero matrix
Every column is free, rank 0
RREF / solution
[0, 0; 0, 0]
Frequently asked questions
What is the difference between REF and RREF?+
Row echelon form only needs a staircase of leading entries with zeros below them. Reduced row echelon form also makes each leading entry 1 and clears the entries above it. RREF is unique; a matrix has many REFs.
How do I know how many solutions a system has from its RREF?+
If a row reads 0 = nonzero, there is no solution. Otherwise, if every variable column has a pivot, the solution is unique; if some column has no pivot, that variable is free and there are infinitely many solutions.
What is a pivot column?+
A column that contains a leading 1 in the RREF. Pivot columns of the original matrix form a basis for its column space, and their count is the rank.
What is a free variable?+
A variable whose column has no pivot. It can take any value, and the pivot variables are then expressed in terms of it. Each free variable adds one parameter to the solution.
Does the order of row operations matter?+
Different orders give different intermediate matrices but the same final RREF, because RREF is unique. The calculator uses the standard forward-then-backward Gauss-Jordan order.
Can I use decimals or fractions?+
Yes. Everything is computed with exact fractions, so 0.1 is treated as 1/10 and no rounding error creeps into the pivots.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026