Linear Algebra Intermediate

Matrix Rank Calculator

Determine the rank of a matrix via row reduction to row echelon form. Count non-zero pivot rows, compute nullity, and see every row operation explained clearly.

Subject: Linear Algebra
Input: Up to 3×4 matrix
Output: rank, nullity, REF
Live
Matrix Input
Result
Enter matrix values and click Calculate

Step-by-Step Work

What Rank Means
rank(A) = number of pivot rows in REF

The rank of a matrix equals the dimension of its column space — how many linearly independent columns (or rows) it has. A matrix has full rank if rank = min(rows, cols).

The Rank-Nullity theorem states: rank(A) + nullity(A) = n (number of columns).

Exam tip — a square matrix is invertible if and only if it has full rank (rank = n).
Interpretation
  • rank = dim(col space) = dim(row space)
  • nullity = dim(null space) = # free variables
  • rank + nullity = # columns (always)
  • Full rank ⟹ only trivial null space
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