Reduced Row Echelon Form (RREF) Solver

Compute the Reduced Row Echelon Form¹ of a matrix using Gauss-Jordan elimination² with detailed step-by-step output.
Useful for solving systems of linear equations, finding matrix rank, and determining linear independence.

A matrix is in Reduced Row Echelon Form (RREF) if it satisfies:

All zero rows are at the bottom
The first non-zero entry in each row (pivot) is 1
Each pivot is to the right of the pivot in the row above (staircase pattern)³
Each pivot is the only non-zero entry in its column

Uses:

Solving systems of linear equations \(Ax = b\)
Finding the rank of a matrix
Determining linear independence of vectors
Finding basis for row space and null space⁴

Rows: Columns:

Display values as: Fraction Decimal

Footnotes

Reduced Row Echelon Form (RREF) is the unique canonical form of a matrix obtained by Gauss-Jordan elimination. Unlike row echelon form (REF), RREF is unique for each matrix — two row-equivalent matrices always share the same RREF. ↑
Gauss-Jordan elimination extends Gaussian elimination by also eliminating entries above each pivot (back-substitution), producing a matrix where each pivot column contains exactly one non-zero entry equal to 1. ↑
The staircase pattern means the pivot positions form a strictly increasing sequence of column indices as you move down the rows. This ensures the echelon structure. ↑
The null space (kernel) of \(A\) is the set of all solutions to \(Ax = 0\). From the RREF, free variables correspond to non-pivot columns, and the null space basis vectors can be read directly. ↑

Cite this tool

Kapita, S. (2026). Reduced Row Echelon Form (RREF) Solver. Math Tools. https://shelvean.github.io/math-tools/rref.html

Kapita, Shelvean. "Reduced Row Echelon Form (RREF) Solver." Math Tools, 2026, shelvean.github.io/math-tools/rref.html.

@online{kapita2026rref,
  author       = {Shelvean Kapita},
  title        = {{Reduced Row Echelon Form (RREF) Solver}},
  year         = {2026},
  organization = {Math Tools},
  url          = {https://shelvean.github.io/math-tools/rref.html}
}