Row Echelon Form (REF) Solver

Compute the Row Echelon Form¹ using Gaussian elimination² with step-by-step output.
For the fully reduced version, see RREF Solver.

A matrix is in Row Echelon Form (REF) if:

All zero rows are at the bottom
The first non-zero entry in each row (pivot) is to the right of the pivot in the row above

REF vs RREF:³

REF only requires forward elimination (Gaussian elimination)
RREF additionally requires back-substitution (Gauss-Jordan elimination)
RREF has pivots = 1 and zeros above each pivot; REF does not

Uses:

Solving systems via back-substitution⁴
Computing determinants (product of pivots times sign from swaps)
Determining rank and linear independence

Rows: Columns:

Enter dimensions and click Generate Matrix to create the input grid.

Display values as: Fraction Decimal

1 — Compute

Performs Gaussian elimination with partial pivoting to row echelon form.

Resets all dimensions, matrix entries, and results.

Footnotes

Row Echelon Form (REF) is an upper-triangular form achieved by forward elimination. Unlike RREF, REF is not unique — different sequences of row operations can yield different echelon forms for the same matrix. ↑
Gaussian elimination is the process of using elementary row operations (row swaps, row scaling, adding a multiple of one row to another) to transform a matrix into row echelon form. It is named after Carl Friedrich Gauss. ↑
REF vs RREF: REF requires only forward elimination and allows non-zero entries above pivots. RREF (Gauss-Jordan) additionally performs back-substitution so each pivot is 1 and is the only non-zero entry in its column. RREF is unique; REF is not. ↑
Back-substitution is the process of solving a system in row echelon form from the bottom row upward. Starting from the last non-zero row, each variable is solved in terms of the variables already determined. ↑

Cite this tool

DOI: https://doi.org/10.5281/zenodo.20981356

Kapita, S. (2026). Row Echelon Form (REF) Solver. Math Tools. https://doi.org/10.5281/zenodo.20981356

Kapita, Shelvean. "Row Echelon Form (REF) Solver." Math Tools, 2026, doi.org/10.5281/zenodo.20981356.

@online{kapita2026refsolver,
  author       = {Shelvean Kapita},
  title        = {{Row Echelon Form (REF) Solver}},
  year         = {2026},
  organization = {Math Tools},
  doi          = {10.5281/zenodo.20981356},
  url          = {https://doi.org/10.5281/zenodo.20981356}
}