Matrix Rank Verifier

Math Linear Algebra • Determinants and Rank

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Find the rank of a matrix by computing its RREF (reduced row echelon form). The rank equals the number of pivot columns (equivalently, the number of non-zero rows in RREF). Also reports nullity using \(\text{rank}+\text{nullity}=\text{#cols}\).

Rows (m)

Columns (n)

Display decimals

Zero tolerance

Steps

Canvas view

Accepted entries: -3.5, 2e-4, fractions like 7/3, and constants pi, e. Pivot columns are highlighted; use Play to animate elimination snapshots.

Matrix \(A\)

2×3

Results

Rank

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Nullity

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Pivot columns

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Full rank?

—

Ready

Pivot structure

Heatmap of the current snapshot. Pivot columns are highlighted; pivot cell is red.

pivot column pivot row pivot cell near-zero row

Drag to pan • wheel to zoom • double-click to reset view

Enter a matrix and click “Calculate”.

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Theory

Rank and what it measures

The rank of a matrix \(A\in\mathbb{R}^{m\times n}\) is the dimension of its column space (and also its row space). Equivalently, it is the maximum number of linearly independent columns (or rows).

Column rank = number of linearly independent columns.
Row rank = number of linearly independent rows.
A key theorem says row rank = column rank; both equal \(\mathrm{rank}(A)\).

RREF and pivot columns

The calculator finds rank by reducing \(A\) to RREF (reduced row echelon form) using Gauss–Jordan elimination. In RREF:

Each non-zero row has a leading 1 (a pivot).
Pivot columns have zeros everywhere except at the pivot.
The number of pivots equals the number of non-zero rows.

Therefore, \[ \mathrm{rank}(A)=\#(\text{pivots})=\#(\text{non-zero rows in RREF}). \]

The pivot columns (in the original matrix) correspond to a set of columns that form a basis for the column space, so they represent a concrete independence test.

Nullity and rank–nullity

The null space of \(A\) is the set of vectors \(x\in\mathbb{R}^n\) such that \(Ax=0\). Its dimension is the nullity.

The rank–nullity theorem states: \[ \mathrm{rank}(A) + \mathrm{nullity}(A) = n, \] where \(n\) is the number of columns. The calculator reports nullity as \(n-\mathrm{rank}(A)\).

Numerical rank (university note)

In exact arithmetic (integers/rationals), pivots are either zero or not. With floating-point data, very small pivots can appear due to rounding or noisy measurements. A tolerance treats values with \(|v|\le \varepsilon\) as zero, producing a practical “numerical rank”.

If your matrix comes from measured data, try adjusting the tolerance to see how stable the rank is.

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