RREF calculator
Calculator
The calculator needs JavaScript. The tables below give the same numbers for common cases.
How it is worked out
- Divide row 1 by 2 so the pivot is 1.
- Subtract -3 × row 1 from row 2.
- Subtract -2 × row 1 from row 3.
- Divide row 2 by 1/2 so the pivot is 1.
- Subtract 1/2 × row 2 from row 1.
- Subtract 2 × row 2 from row 3.
- Divide row 3 by -1 so the pivot is 1.
- Subtract -1 × row 3 from row 1.
- Subtract 1 × row 3 from row 2.
| Inputs | Answer |
|---|---|
| Matrix: 2 1 -1 8; -3 -1 2 -11; -2 1 2 -3 | The reduced row echelon form has 3 pivots (rank 3): 1 0 0 2 | 0 1 0 3 | 0 0 1 -1 (rows separated by |). |
| Matrix: 1 2 3; 4 5 6; 7 8 9 | The reduced row echelon form has 2 pivots (rank 2): 1 0 -1 | 0 1 2 | 0 0 0 (rows separated by |). |
| Matrix: 1 1 5; 1 -1 1 | The reduced row echelon form has 2 pivots (rank 2): 1 0 3 | 0 1 2 (rows separated by |). |
How to use it
Reduced row echelon form (RREF) is the result of Gauss-Jordan elimination: each leading entry is 1, it is the only non-zero entry in its column, and the leading entries move to the right as you go down. For a system of linear equations written as an augmented matrix, the RREF shows the solution directly.
The calculator works in exact fractions so there is no rounding, lists the row operations it used, and reports the rank and the pivot columns. Enter an augmented matrix such as 2 1 −1 8; −3 −1 2 −11; −2 1 2 −3 to solve a 3 × 3 system.
Frequently asked questions
What is reduced row echelon form?
A matrix with leading 1s, zeros above and below each leading 1, and leading 1s stepping to the right.
How do I solve equations with RREF?
Write the augmented matrix and row-reduce; the last column gives the solution when the left side becomes the identity.
What is the rank of a matrix?
The number of pivots (non-zero rows) in its row echelon form.
Method: standard arithmetic shown step by step; no outside data is used. Checked 2026-11-12.
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