Matrix Rank Calculator

Free Client-Side Step-by-Step

Calculate matrix rank with step-by-step row echelon form (REF) transformation. Shows rank(A), nullity, pivot positions, and linearly independent rows for m×n matrices. 100% client-side—no data sent to servers. Supports 2×2 up to 8×8.

Matrix Input
One row per line, space separated
Quick Examples
Result
Enter a matrix and click "Calculate Rank" to see the result.
Step-by-Step Solution
Detailed row reduction steps will appear here.
About Matrix Rank

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 row echelon form.

Properties:

  • rank(A) ≤ min(rows, columns)
  • Full rank means rank equals the smaller dimension
  • Rank deficient means rank is less than the smaller dimension
  • Zero matrix has rank 0

Applications:

  • Linear Systems: Determines if Ax = b has solutions
  • Nullity: nullity = n − rank (dimension of null space)
  • Invertibility: Square matrix is invertible iff rank = n
  • Span: Rank tells dimension of column/row space

Exam-Style Practice

About This Matrix Rank Calculator & Methodology

The rank of a matrix is the maximum number of linearly independent rows (or columns). This tool uses Gaussian elimination to reduce to row echelon form (REF), then counts non-zero pivot rows. The rank-nullity theorem: rank(A) + nullity(A) = n. All calculations run client-side—no data stored.

Authorship & Expertise

  • Author: Anish Nath
  • Background: Math and developer tools for education
  • Method: Gaussian elimination to REF

Trust & Privacy

  • Privacy: All calculations run locally; no data stored
  • Client-side: Your matrices never leave your device
  • Support: @anish2good

Matrix Rank: FAQ

How do I find the rank of a matrix?

Enter your matrix and click Calculate. The tool reduces it to row echelon form (REF) and counts non-zero rows (pivot rows) to obtain rank(A).

What is nullity and how is it related to rank?

Nullity is the dimension of the null space. For an m×n matrix, the rank-nullity theorem states rank(A) + nullity(A) = n.

What does rank tell me about a matrix?

Full rank means maximum linearly independent columns. If rank(A) < n for an n×n square matrix, A is singular and not invertible.

What is the rank-nullity theorem?

For an m×n matrix A, rank(A) + nullity(A) = n. This means the number of pivot columns plus the number of free variables always equals the total number of columns. It connects the column space and null space dimensions.

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