Matrix Rank Calculator

Use our free matrix rank calculator to find the rank of any matrix, view the RREF, and determine linear dependence of rows and columns.

Math

Mode

Matrix

Formula

The rank of a matrix is the number of linearly independent rows (or columns). It's computed by converting to reduced row echelon form (RREF) via Gaussian elimination and counting non-zero rows.

Rank = number of non-zero rows in RREF

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Select a mode.
2Enter any matrix (rectangular or square).
3Rank: directly shows rank and full-rank status.
4Steps: shows row operations.
5Dependence: shows independence analysis.
6Click Calculate.

Examples

3×3 full rank

1.RREF = identity matrix
2.Rank = 3

Result: 3

3×3 rank 2

1.Row 3 is dependent on rows 1 and 2
2.Rank = 2

Result: 2

Frequently Asked Questions

The dimension of the column space (or row space).

min(rows, columns). Full rank = rank = min(rows, cols).

Reduced Row Echelon Form — each pivot is 1 with zeros above and below.

rank + nullity = number of columns.

Determines solvability of linear systems (Ax = b has a solution iff rank(A) = rank([A|b])).

Related Calculators

Matrix Calculator Matrix Determinant Calculator Matrix Inverse Calculator Matrix Transpose Calculator Percentage Calculator Basic Calculator

CalculateMe Team

Last updated: 2026-07-14

Matrix Rank Calculator

Formula

How It Works

Examples

3×3 full rank

3×3 rank 2

Frequently Asked Questions

What is rank?

Max rank?

What is RREF?

Rank-nullity theorem?

Why is rank useful?

Related Calculators

Matrix Rank Calculator

Formula

How It Works

Examples

3×3 full rank

3×3 rank 2

Frequently Asked Questions

What is rank?

Max rank?

What is RREF?

Rank-nullity theorem?

Why is rank useful?

Related Calculators