About the RREF Calculator
This RREF calculator puts any matrix up to 8 × 9 into reduced row echelon form using Gauss–Jordan elimination, and lists every elementary row operation it performs — row swaps, scaling a row by its pivot, and subtracting multiples of one row from another. All arithmetic is done with exact fractions, so you see 1/3 instead of 0.333333 and never get rounding errors.
It is for linear algebra students checking homework, and for anyone solving a system of linear equations, finding the rank of a matrix, identifying pivot and free columns, or testing whether vectors are linearly independent. Type one row per line, with entries separated by spaces or commas; fractions such as 3/4 and decimals are accepted.
Switch on "Last column is the constants" when you enter an augmented matrix [A | b] and the calculator will also say whether the system has a unique solution, infinitely many solutions or none, and give the unique solution when there is one.
How to use the rref calculator
- 1Type the matrix with one row per line and entries separated by spaces or commas.
- 2Switch on "Last column is the constants" if it is an augmented matrix for a linear system.
- 3Read the reduced row echelon form, shown with exact fractions.
- 4Check the rank, pivot columns and free columns.
- 5Follow the numbered row operations to reproduce the working by hand.
Formula and method
Gauss–Jordan elimination works column by column. In each column it finds the first row (at or below the current pivot row) with a non-zero entry, swaps it into place, divides the row by that entry to make the pivot 1, and then subtracts multiples of the pivot row from every other row to make the rest of the column zero. Columns with no usable entry are skipped and become free columns.
The result is the unique reduced row echelon form: each non-zero row starts with a leading 1, each leading 1 is to the right of the one above, and every leading 1 is the only non-zero entry in its column. The number of pivots is the rank. Different sequences of row operations can be used, but they always end at the same RREF.
- Rᵢ
- Row i of the matrix
- k
- The pivot value used to scale a row
- c
- Multiplier used to clear an entry
- rank
- Number of pivots (leading 1s)
Worked examples
3 × 3 linear system (augmented)
The system x + 2y − z = −4, 2x + 3y − z = −11, −2x − 3z = 22 reduces to the identity on the left with constants −8, 1, −2 on the right, so x = −8, y = 1, z = −2. Substituting back: −8 + 2 + 2 = −4 ✓.
Singular 3 × 3 matrix
Row 3 is 2 × row 2 − row 1, so one row vanishes. The RREF has pivots in columns 1 and 2, rank 2, and column 3 is free: it equals −1 × column 1 + 2 × column 2.
Infinitely many solutions
The second equation is twice the first, so only two independent equations remain for three unknowns. Solving gives x = y = 3 − z/2 with z free — for example z = 2 gives (2, 2, 2).
Inconsistent system with fractions
Doubling the first row gives x + 2y = 6, but the second row says x + 2y = 5. Elimination leaves the row [0 0 | 1], meaning 0 = 1, so there is no solution.
Frequently asked questions
What is reduced row echelon form?+
A matrix is in reduced row echelon form (RREF) when all zero rows are at the bottom, each non-zero row starts with a leading 1, each leading 1 sits to the right of the one above it, and each leading 1 is the only non-zero number in its column.
What is the difference between REF and RREF?+
Row echelon form (REF) only needs zeros below each pivot, and pivots need not be 1. RREF additionally requires every pivot to be 1 and zeros above the pivots too. REF is not unique, but every matrix has exactly one RREF.
How do I tell if a system has no solution from the RREF?+
Look for a row whose coefficient entries are all zero but whose last (constant) entry is non-zero, i.e. [0 0 … 0 | 1]. That row says 0 = 1, so the system is inconsistent. If there is no such row and there are free variables, there are infinitely many solutions.
How is rank found from the RREF?+
The rank equals the number of leading 1s (pivots) in the reduced row echelon form, which is also the number of non-zero rows. It tells you how many rows or columns are linearly independent.
Why does the calculator use fractions?+
Row reduction repeatedly divides, and decimals like 0.333… introduce rounding errors that can make a zero look non-zero. Exact rational arithmetic keeps every entry precise, which matters for deciding rank and consistency correctly.