RREF Calculator (Reduced Row Echelon Form)
Reduce any matrix to reduced row echelon form with step-by-step Gauss-Jordan elimination. Free RREF calculator that shows every row operation, the rank and supports fractions.
Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser
Tip: for a system Ax = b, append b as the last column (an augmented matrix). Fractions like 3/4 are allowed.
Show the row operations
How the RREF Calculator Works
This tool reduces any matrix to its reduced row echelon form using Gauss-Jordan elimination, showing every row operation along the way. Enter one row per line, with numbers separated by spaces or commas. You can enter fractions like 3/4, and for a linear system you append the constants as the final column to form an augmented matrix.
The Formula
A matrix is in reduced row echelon form when all of these hold:
- Each leading nonzero entry (the pivot) in a row equals 1.
- Each pivot is the only nonzero entry in its column.
- The pivot in any row is to the right of the pivot in the row above it.
- All-zero rows are at the bottom.
The algorithm scans columns left to right. For each column it selects a pivot (using partial pivoting for numerical stability), scales the pivot row so the pivot becomes 1, then eliminates that column in every other row.
Worked Example
Solve this system written as an augmented matrix:
| 1 2 -1 | -4 | | 2 3 -1 | -11 | | -2 0 -3 | 22 |
After Gauss-Jordan elimination the RREF is:
| Variable | Pivot row | Value |
|---|---|---|
| x | 1 0 0 | -8 |
| y | 0 1 0 | 1 |
| z | 0 0 1 | -2 |
So x = -8, y = 1, z = -2, and the rank is 3 — one pivot per variable, meaning the system has exactly one solution.
Reading the Result
- A full set of pivots (rank = number of variables) means a unique solution.
- A row like
0 0 0 | 5(all zeros, nonzero constant) signals an inconsistent system with no solution. - Fewer pivots than variables means infinitely many solutions, with free variables corresponding to non-pivot columns.
Why RREF Matters
Reduced row echelon form is the backbone of linear algebra. It lets you solve systems of equations, find the rank of a matrix, determine whether vectors are linearly independent, compute matrix inverses (reduce [A | I]), and describe the null space and column space of a transformation.
Frequently asked questions
What is reduced row echelon form (RREF)?+
RREF is a unique standardized form of a matrix where each leading nonzero entry (pivot) equals 1, each pivot is the only nonzero number in its column, every pivot sits to the right of the one above it, and any all-zero rows are at the bottom. It is reached using Gauss-Jordan elimination.
How does Gauss-Jordan elimination work?+
Pick the leftmost column with a nonzero entry, move that row up, and divide it so the pivot becomes 1. Then subtract multiples of that row from every other row to make the rest of the column zero. Repeat with the next column and row until the matrix is fully reduced.
What is the difference between REF and RREF?+
Row echelon form (REF) only requires zeros below each pivot, so pivots need not be 1 and entries above pivots can be nonzero. Reduced row echelon form (RREF) goes further: every pivot is 1 and is the only nonzero entry in its column, making the solution readable directly.
How do I solve a system of equations with RREF?+
Write the system as an augmented matrix with the constants as the last column, then reduce to RREF. Each pivot row then reads as a single variable equal to a value. A row like 0 0 1 | -2 means that variable equals -2.
What does the rank of a matrix tell me?+
The rank is the number of pivots, which equals the number of nonzero rows in the RREF. It tells you how many independent equations or rows you have. For a square system, a rank equal to the number of variables means there is exactly one solution.