RREF Calculator
Enter any matrix up to 6×6 and this calculator reduces it to Reduced Row Echelon Form (RREF) using Gauss-Jordan elimination — showing every row operation along the way, the final pivot positions, and the matrix rank. Use it to solve systems of linear equations, check linear independence, or find a matrix inverse.
What Is RREF?
Reduced Row Echelon Form (RREF) is a unique, fully simplified version of any matrix, reached by applying a sequence of elementary row operations. A matrix is in RREF when it satisfies four conditions:
- Every nonzero row has a leading 1 (called a pivot) as its first nonzero entry
- Each pivot is the only nonzero entry in its entire column
- Each pivot sits strictly to the right of the pivot in the row above it (a staircase pattern)
- Any rows that are entirely zero are pushed to the bottom of the matrix
Unlike ordinary row echelon form, RREF is completely unique — no matter which sequence of row operations you use to get there, every matrix reduces to exactly one RREF. That uniqueness is what makes it so useful for reading off answers directly, rather than doing further back-substitution.
Row Echelon Form vs Reduced Row Echelon Form
| Requirement | Row Echelon Form (REF) | Reduced Row Echelon Form (RREF) |
|---|---|---|
| Pivots move right going down rows | Yes | Yes |
| Entries below each pivot are zero | Yes | Yes |
| Each pivot equals exactly 1 | Not required | Required |
| Entries above each pivot are zero | Not required | Required |
| Uniqueness | Many valid forms | Exactly one result |
REF is what you get after forward (Gaussian) elimination alone; RREF is what you get after also clearing out everything above each pivot (the "Jordan" half of Gauss-Jordan elimination).
The Gauss-Jordan Elimination Algorithm
This calculator (and the standard by-hand method) reduces a matrix using only three elementary row operations, which never change the underlying solution set:
- Row swap — exchange the positions of two rows
- Row scaling — multiply every entry in a row by a nonzero constant
- Row addition — add a multiple of one row to another row
The algorithm works column by column, left to right: find a nonzero entry in the current column (swapping rows if needed), scale that row so the entry becomes 1 (the pivot), then use row addition to zero out every other entry in that column — both below and above the pivot. Moving to the next column and repeating eventually produces the full RREF.
Solving Linear Systems from RREF
To solve a system of linear equations, write it as an augmented matrix — the coefficients of each equation, plus a final column for the constants on the right-hand side — then reduce the whole thing to RREF.
- Unique solution: every variable column has a pivot; read each variable's value directly from the augmented column of its pivot row
- No solution: a row reads all zeros across the variable columns but a nonzero value in the augmented column — this represents a contradiction like 0 = 5
- Infinitely many solutions: there are fewer pivots than variables; the non-pivot variables are "free" and the pivot variables can be expressed in terms of them
Matrix Rank
The rank of a matrix is the number of linearly independent rows it contains, and RREF makes it trivial to read: rank equals the number of nonzero rows (equivalently, the number of pivot columns) in the RREF. A matrix has "full rank" when its rank equals the smaller of its row count or column count — meaning no row is a combination of the others.
Finding a Matrix Inverse with RREF
To invert an n×n matrix A: build the augmented matrix [A | I] (A next to the identity matrix), then row-reduce the whole thing to RREF. If the left side becomes the identity matrix, the right side is A⁻¹. If the left side cannot reach the identity (fewer than n pivots), A has no inverse (it is singular).
Worked Example
Consider the system: x + 2y − z = 3, 2x − y + z = 1, and 3x + y − 2z = 4. As an augmented matrix:
Enter these values into the calculator above (3 rows, 4 columns) and press "Reduce to RREF" to see every elimination step and the final solution for x, y, and z.
Glossary
- Pivot
- The first nonzero entry in a row of a matrix in echelon form; in RREF, every pivot equals exactly 1.
- Augmented Matrix
- A matrix formed by combining a coefficient matrix with an extra column (or columns) representing constants, typically used to represent a system of linear equations.
- Rank
- The number of linearly independent rows (or columns) in a matrix, equal to the number of pivots in its RREF.
- Free Variable
- A variable in a linear system whose column in the RREF has no pivot, meaning it can take any value while the pivot variables adjust accordingly.
- Singular Matrix
- A square matrix that has no inverse, because its RREF cannot be reduced to the identity matrix (it has fewer than n pivots).
- Elementary Row Operation
- One of three operations — row swap, row scaling, or row addition — used to transform a matrix without changing the solution set it represents.
Frequently Asked Questions
Q: What is RREF (Reduced Row Echelon Form)?
A: A unique simplified matrix form where every pivot equals 1, is the only nonzero entry in its column, pivots move right down the rows, and zero rows sit at the bottom.
Q: What is the difference between row echelon form and reduced row echelon form?
A: REF only requires pivots below-left cleared to zero; RREF additionally requires pivots to equal 1 and clears entries above pivots too.
Q: How do you solve a system of linear equations using RREF?
A: Build an augmented matrix, reduce to RREF, then read each pivot variable's value from the augmented column of its row.
Q: What is the rank of a matrix and how does RREF show it?
A: Rank is the number of linearly independent rows, equal to the number of nonzero rows (pivots) in the RREF.
Q: What are elementary row operations?
A: Row swap, row scaling by a nonzero constant, and adding a multiple of one row to another.
Q: Can RREF be used to find a matrix inverse?
A: Yes — reduce [A | I] to RREF; if the left side becomes the identity, the right side is A⁻¹.