Matrix Rank Calculator – Find Rank of a Matrix

Calculate the rank of any matrix with our free online matrix rank calculator. Get step-by-step row reduction to echelon form with detailed explanations.

Rows:

Columns:

Examples:

Enter matrix values (each row on a new line, values separated by spaces or commas)

Current matrix: 3x3 | Detected from your input

Understanding Matrix Rank

The rank of a matrix tells you how many rows (or columns) are truly independent – not expressible as combinations of others. It's a measure of the "information content" or dimensionality of the matrix. A full-rank matrix has maximum possible rank; a rank-deficient matrix has redundant rows or columns.

Rank is found by reducing the matrix to row echelon form using Gaussian elimination. The rank equals the number of non-zero rows in this reduced form – each represents an independent piece of information.

Key Concepts

Row Rank

The number of linearly independent rows. Found by row reduction to echelon form.

Column Rank

The number of linearly independent columns. Always equals row rank!

Full Rank vs Rank Deficient

For an m x n matrix, maximum possible rank is min(m, n).

Full rank: rank = min(m, n) – maximum independence
Rank deficient: rank < min(m, n) – some redundancy exists
Zero matrix: rank = 0 – complete redundancy

Worked Examples

Example 1: Identity Matrix

I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Already in echelon form with 3 non-zero rows

Rank = 3 (full rank for 3x3)

Example 2: Singular Matrix

A = [[1, 2, 3], [2, 4, 6], [3, 6, 9]]

Row 2 = 2 x Row 1, Row 3 = 3 x Row 1

After reduction: only 1 non-zero row remains

Rank = 1 (highly rank deficient)

Example 3: Rectangular Matrix

A = [[1, 2, 3, 4], [5, 6, 7, 8]] (2x4 matrix)

Maximum possible rank = min(2, 4) = 2

Rows are independent (not multiples)

Rank = 2 (full row rank)

Example 4: Zero Matrix

0 = [[0, 0], [0, 0]]

All rows are zero – complete redundancy

Rank = 0

Quick Fact

The rank-nullity theorem is one of the most important results in linear algebra. It states that for any matrix, rank + nullity = number of columns. The nullity tells you the dimension of the solution space for Ax = 0. This theorem connects the matrix's structure to the solutions of linear systems.

Frequently Asked Questions

What does rank tell me about a system of equations?

For Ax = b, if rank(A) equals the number of variables, there's a unique solution. If rank is less, there are either no solutions or infinitely many. Rank tells you how many independent constraints you have.

Can rank be greater than the number of rows?

No. Rank is bounded by both dimensions: rank ≤ min(rows, columns). A 3x5 matrix can have rank at most 3. A 5x3 matrix can have rank at most 3.

Is row rank always equal to column rank?

Yes! This is a fundamental theorem of linear algebra. The number of independent rows always equals the number of independent columns, even for rectangular matrices.

How does rank relate to invertibility?

A square n x n matrix is invertible if and only if it has full rank (rank = n). Rank-deficient square matrices are singular and have no inverse.

What is nullity?

Nullity is the dimension of the null space – all vectors x where Ax = 0. By the rank-nullity theorem: nullity = columns - rank. It tells you how many "free variables" exist in the homogeneous system.

Where is matrix rank used in practice?

Rank appears in control theory (system controllability), statistics (multicollinearity detection), machine learning (feature independence), computer vision (structure from motion), and network analysis (connectivity).

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