Matrix RREF Calculator
Reduce any matrix to Row Reduced Echelon Form (RREF) using Gauss-Jordan elimination. Shows pivot positions, rank, and step-by-step row operations.
What Is the Matrix RREF Calculator?
Row Reduced Echelon Form (RREF) is the unique canonical form of a matrix obtained by Gauss-Jordan elimination. In RREF, each leading nonzero entry (pivot) equals 1, each pivot column has zeros in every other row, and pivots appear strictly to the right of pivots in earlier rows. RREF directly reveals the rank, null space, and solutions of a linear system.
Formula
How to Use
Select the matrix dimensions (rows × columns, up to 5×6 for augmented systems). Enter the matrix entries. For a linear system Ax=b, enter the augmented matrix [A|b] by adding a column for b. Click Reduce to RREF. The pivot positions and rank are highlighted.
Example Calculation
[1 2 3; 4 5 6; 7 8 9] reduces to RREF with rank 2, showing the system is linearly dependent (rows 2 and 3 are linear combinations of row 1).
Understanding Matrix RREF
Gauss-Jordan elimination is the systematic algorithm for reducing a matrix to RREF. It extends Gaussian elimination (which produces REF) by continuing to eliminate entries above each pivot as well as below.
The algorithm visits each pivot position from left to right, top to bottom: scale the pivot row so the pivot = 1, then subtract multiples of that row from all other rows to zero out the pivot column. The result is the unique RREF.
Numerical stability is important for floating-point implementations: partial pivoting (choosing the largest entry in a column as the pivot) reduces round-off errors. This calculator uses partial pivoting with a small tolerance (10^−12) to identify near-zero entries as exactly zero.
RREF is the foundation of computational linear algebra. Virtually every linear algebra subroutine — LU decomposition, QR factorisation, Gram-Schmidt orthogonalisation — is related to systematic row operations.
Frequently Asked Questions
What is RREF used for?
RREF is used to solve systems of linear equations, find the rank and null space of a matrix, determine linear independence of vectors, and compute matrix inverses (using the augmented matrix [A|I] approach).
What is the difference between REF and RREF?
Row Echelon Form (REF) has each pivot in a staircase pattern but pivots may not equal 1 and columns above pivots may have nonzeros. RREF additionally requires pivots = 1 and that columns above each pivot are also zeroed out. RREF is unique; REF is not.
How do you read the solution from RREF?
For an augmented matrix [A|b], pivot columns correspond to basic variables and non-pivot columns to free variables. Each row with a pivot gives an equation expressing that basic variable in terms of the free variables and the rightmost column.
What does it mean if RREF has a row [0 0 … 0 | c] where c ≠ 0?
This indicates an inconsistent system — no solution exists. For example, the row [0 0 1] in a 3-column augmented matrix represents 0·x + 0·y = 1, which is impossible.