rk Matrix Rank Calculator

Matrix Rank Calculator

Find the rank of any matrix using Gaussian elimination. 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 SymPy 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