Matrix Rank Calculator

Rank via row reduction

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About Matrix Rank Calculator

A matrix rank calculator using Gaussian elimination to row echelon form. Shows pivot positions, rank, nullity, and determines system solvability. Handles rectangular matrices. Select from preset matrices. All calculations are client-side. Essential for understanding linear system behavior.

Matrix Rank Calculator Features

  • Rank
  • Nullity
  • Pivots
  • REF
  • Presets
Matrix rank: the number of linearly independent rows (or columns). Equals the number of pivots in row echelon form. rank + nullity = n (number of columns). Determines solution count: rank(A)=rank([A|b]) for consistency; free vars = n − rank.

How to Use

Enter a matrix:

  • A: Any m×n matrix
  • Output: Rank, nullity, REF
  • Pivots: Highlighted positions

Rank-Nullity

rank(A) + nullity(A) = n. This fundamental theorem connects the dimension of the column space (rank) to the dimension of the null space (nullity). Always holds for any matrix.

Applications

  • System solvability
  • Linear independence test
  • Dimension of subspaces
  • Controllability/observability

Step-by-Step Instructions

  1. 1Enter a matrix.
  2. 2View row echelon form.
  3. 3Count pivots (= rank).
  4. 4Compute nullity.
  5. 5Determine solvability.

Matrix Rank Calculator — Frequently Asked Questions

What does rank tell us about Ax=b?+

If rank(A) = rank([A|b]) = n: unique solution. If rank(A) = rank([A|b]) < n: infinite solutions (n−rank free variables). If rank(A) < rank([A|b]): no solution (inconsistent).

Is row rank equal to column rank?+

Yes, always. This is a fundamental theorem of linear algebra. The dimension of the row space equals the dimension of the column space for any matrix. Both equal the number of pivots.

How does rank relate to invertibility?+

An n×n matrix is invertible iff rank = n (full rank). Equivalent to: det ≠ 0, null space = {0}, columns linearly independent, all eigenvalues nonzero.

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