In reduced row echelon form (RREF) every pivot is 1, lies right of the pivot above, and is the only nonzero entry in its column. The augmented matrix [2 1 −1 8; −3 −1 2 −11; −2 1 2 −3] reduces to [1 0 0 2; 0 1 0 3; 0 0 1 −1], so x = 2, y = 3, z = −1.
Students checking homework, engineers solving small linear systems and anyone verifying a hand reduction can paste a matrix here and see its reduced row echelon form. Type one row per line with entries separated by spaces or commas; fractions like 3/4 are read exactly as written. The result gives the rank, which columns hold pivots, how many columns are free, and the reduced rows shown as fractions where a small denominator fits. The working lists each row exchange, scaling and elimination in order. The arithmetic is floating point with partial pivoting, so entries that should be exactly zero are cleaned below 1e-10; for matrices with very large or very small entries in the same column, treat a borderline rank with care.
Put each row on its own line and separate entries with spaces or commas. Decimals and fractions such as -5/3 both work.
The headline number is the rank. Pivot columns tell you which variables are basic; non-pivot columns are free variables.
The table shows each row of the RREF. For an augmented system, the last column holds the solution when every variable column has a pivot.
The steps list every row operation, such as R2 = R2 − (−3/2) × R1, so you can compare against your own reduction.
Result Rank: 3
Matrix size (rows × columns): 3 × 4
Pivot columns: 1, 2, 3
Non-pivot columns: 1
RREF (rows separated by |): 1 0 0 2 | 0 1 0 3 | 0 0 1 -1
The augmented matrix of three equations in x, y and z. Its last column after reduction reads off the solution x = 2, y = 3, z = −1.
Result Rank: 2
Matrix size (rows × columns): 3 × 3
Pivot columns: 1, 2
Non-pivot columns: 1
RREF (rows separated by |): 1 0 -1 | 0 1 2 | 0 0 0
Row 3 equals twice row 2 minus row 1, so one row vanishes and the rank drops to 2.
Gauss-Jordan: for each column, choose the largest pivot, scale its row to 1, then subtract multiples to clear the column above and below
RᵢprankThe tool walks the columns left to right. In each column it picks the row with the largest absolute value at or below the current row (partial pivoting, which limits rounding error), swaps it up, divides it by the pivot, and subtracts that row from every other row to zero the column. A column with no usable pivot is skipped and becomes a free column. The RREF of a matrix is unique, so any correct sequence of operations ends at the same matrix.
Row echelon form only requires each pivot to sit to the right of the one above, with zeros below it. Reduced row echelon form adds two conditions: every pivot equals 1 and every other entry in a pivot column is 0. A matrix has many row echelon forms but exactly one RREF.
Reduce the augmented matrix. If any row reads 0 0 … 0 | c with c not zero, the equations contradict each other and the system is inconsistent. Equivalently, the pivot list includes the last column.
In a coefficient or augmented matrix, each non-pivot variable column is a free variable that can take any value, so a consistent system with free columns has infinitely many solutions. The count also gives the dimension of the null space: columns minus rank.
Yes. Write entries like 2/3 or -7/4. Results are shown as fractions when the value matches a denominator up to 12, and as decimals to four places otherwise.
The tool uses partial pivoting: it moves the largest entry in the column into the pivot position to reduce rounding error. Textbooks often pick the first nonzero entry instead. The intermediate steps differ, but the final RREF is the same because it is unique.
A square n × n matrix is invertible exactly when its rank is n, which means its RREF is the identity matrix. A 3 × 3 matrix with rank 2, such as the 1-to-9 matrix, is singular and has determinant 0.
Using this tool in an assignment, article or report? Copy a citation, or a link for your website.
Tools Directory Editorial Team. (2026, October 24). RREF Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/algebra-tools/rref-calculator/
Tools Directory Editorial Team. “RREF Calculator.” Tools Directory, 24 Oct. 2026, https://tools.milunir.com/math/algebra-tools/rref-calculator/. Accessed 26 Oct. 2026.
Tools Directory Editorial Team. “RREF Calculator.” Tools Directory. Last modified October 24, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/algebra-tools/rref-calculator/.
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