What is Row Reduced Echelon Form (RREF)?

A matrix is in RREF when: (1) all zero rows are at the bottom, (2) the leading entry of each non-zero row (the pivot) is 1, (3) each pivot is the only non-zero entry in its column, and (4) pivots move strictly right as you go down.

How is RREF different from Row Echelon Form (REF)?

REF only requires zeros below each pivot; the pivot does not have to be 1 and can have non-zero entries above it. RREF goes further: it makes every pivot equal to 1 and eliminates entries both above and below each pivot.

Can I use this for a non-square matrix?

Yes. RREF applies to any m×n matrix, not just square ones. This is useful for augmented matrices representing systems with more unknowns than equations, or vice versa.

What does it mean if there is no pivot in a column?

A column without a pivot is a free variable column. In a system of equations this means that variable can take any value, resulting in infinitely many solutions (or no solution if a contradiction row exists).

What row operations does Gauss-Jordan use?

Three elementary row operations: (1) Swap two rows (R_i ↔ R_j), (2) Scale a row by a non-zero constant (R_i → k·R_i), and (3) Replace a row with itself plus a multiple of another row (R_i → R_i + k·R_j). None of these change the solution set of the linear system.

Why does the calculator show fractions as decimals?

All arithmetic is done in floating-point. Values within 1×10⁻⁹ of an integer are snapped to that integer to eliminate rounding noise; other values are shown to up to 6 significant figures.

Matrix RREF Calculator

Enter a matrix, choose its size, and see the full Gauss-Jordan elimination step by step until RREF is reached.

Matrix Input

Rows:

Cols:

Enter matrix values and click Calculate RREF

Summary

This tool reduces a matrix of up to 5 rows and 6 columns to Row Reduced Echelon Form (RREF) using Gauss-Jordan elimination. Enter values, click Calculate, and every row operation — scaling, swapping, and row replacement — is shown in sequence so you can follow each transformation from the original matrix to the final RREF.

How it works

Select the number of rows and columns for your matrix (up to 5×6).
Fill in each cell with a numeric value; decimals and negatives are accepted.
Click "Calculate RREF" to run Gauss-Jordan elimination.
Each row operation (scale, swap, replace) is listed as a step with the resulting matrix.
The final RREF is highlighted at the bottom with pivot positions marked.
Use "Random" to load a sample matrix, or "Clear" to start over.

Use cases

Solve systems of linear equations by reducing the augmented matrix.
Find the rank of a matrix by counting pivot columns in RREF.
Determine whether a system has no solution, one solution, or infinitely many.
Verify homework answers with step-by-step Gauss-Jordan working.
Find the null space / kernel of a matrix.
Check linear independence of a set of vectors.
Compute the inverse of a matrix using an augmented [A | I] setup.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu