Row Echelon Form Reducer

Math Linear Algebra • Matrices and Systems of Equations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Reduce a matrix to REF (row echelon form) or RREF (reduced row echelon form) using elementary row operations. Optionally treat the last column as an augmented \(\mathbf{b}\) column.

Rows (m)

Cols (n)

Output

Display decimals

Pivot tolerance

Steps

Augmented matrix (treat last column as \(\mathbf{b}\)) Show pivot-column basis vectors (from the original matrix)

Play animates the row-reduction snapshots once and stops automatically. Drag to pan • wheel to zoom • double-click to reset.

Input matrix

2×3

REF

—

RREF

—

Ready

Row operation animation

Pivot cell and pivot row are highlighted. If augmented, the last column is highlighted separately.

matrix augmented column pivot pivot row target row

Enter a matrix and click “Calculate”.

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Theory

REF vs RREF

Row reduction transforms a matrix using elementary row operations: swap rows, scale a row by a nonzero constant, or replace a row by “row + multiple of another row”. These operations preserve linear relationships, which is why echelon forms are central to systems, rank, and linear dependence.

A matrix is in row echelon form (REF) when each pivot (leading nonzero entry) steps to the right as you go down, and all entries below each pivot are zero. A matrix is in reduced row echelon form (RREF) when it is in REF and each pivot is 1 and is the only nonzero entry in its column (zeros both below and above).

Pivots, pivot columns, and rank

A pivot is the leading 1 (or leading nonzero entry, before normalization) in a pivot row. The number of pivot rows equals the rank (counting pivots only in the “A-part” columns when the augmented option is enabled).

Pivot columns also identify which columns of the original matrix form a basis for the column space. That is why this calculator can show “pivot-column basis vectors”: it takes the pivot column indices from the reduction, then extracts those columns from the original input matrix and lists them as vectors.

Augmented matrices (optional)

If you enable the augmented option, the last column is treated as a right-hand side \(\mathbf{b}\), giving an augmented matrix \([A\mid \mathbf{b}]\). RREF can reveal inconsistency via a row of the form \([0\ \cdots\ 0\mid c]\) with \(c\neq 0\). This tool warns you when that happens.

For full system solving (unique / infinite / none, and parametric solutions), use your Linear System Solver. This tool is focused on producing clean REF/RREF and showing the row-operation sequence.

Copy buttons

The “Copy REF” and “Copy RREF” buttons copy the corresponding matrix as a JSON-style nested array such as [[1,0,-1],[0,1,2]]. This is useful for quickly pasting results into other calculators or notes.