rk Matrix Rank Calculator

Matrix Rank Calculator

The rank of a matrix is the number of linearly independent rows (or columns), and this calculator finds it by reducing the matrix to row echelon form. Shows RREF, identifies all pivot columns, and computes nullity = cols − rank.

rk

Matrix Rank Calculator

Rank via RREF with pivot identification

✓ Step-by-Step✓ Nullity✓ Free⚡ Loading precision engine…
Matrix A
Quick Examples

What is Matrix Rank?

The rank of a matrix is the maximum number of linearly independent rows (or columns). It equals the number of non-zero rows in the RREF of the matrix.

Rank-Nullity Theorem

$$\text{rank}(A) + \text{nullity}(A) = n \quad \text{(number of columns)}$$

Interpretations

  • Full column rank: rank = n (columns are linearly independent)
  • Full row rank: rank = m (rows are linearly independent)
  • Full rank: rank = min(m, n)
  • rank(A) = rank(Aᵀ) always

Worked Example

For A = [[1,2],[2,4]]: the second row is exactly 2 times the first row, so it adds no new information. This matrix has rank 1, even though it's a 2×2 matrix — because only one row is truly independent.

How It's Calculated

This calculator uses row reduction (RREF) to simplify the matrix, then counts the number of non-zero rows remaining. Each non-zero row after simplification represents one independent direction in the matrix.

Applications

  • Determining whether a system of equations has a unique solution
  • Checking whether a matrix can be inverted (only full-rank square matrices have inverses)
  • Data compression and dimensionality reduction in machine learning

Frequently Asked Questions

A matrix has full rank when its rank equals the smaller of its number of rows or columns — meaning every row or column contributes independent information.
No — the rank can never be larger than the smaller of the number of rows and the number of columns in the matrix.
The rank tells you how many independent equations you actually have. If the rank is lower than the number of unknowns, your system won't have a single unique solution.

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