Matrix Rank Calculator
Calculate the rank of a matrix using Gaussian elimination.
Select Matrix Size
Matrix Rank
The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix.
Properties:
- rank(A) ≤ min(m,n) for an m×n matrix
- rank(A) = rank(A^T)
- rank(AB) ≤ min(rank(A), rank(B))
- rank(A+B) ≤ rank(A) + rank(B)
- rank(A) = n if and only if A is invertible (for n×n matrix)
Methods to Calculate Rank:
- Gaussian Elimination (Row Echelon Form)
- Singular Value Decomposition (SVD)
- QR Decomposition
Applications:
- Solving systems of linear equations
- Determining linear independence
- Finding basis for row/column space
- Matrix factorization
- Image processing and compression
Special Cases:
- Full rank: rank = min(m,n)
- Rank deficient: rank < min(m,n)
- Zero matrix: rank = 0
- Identity matrix: rank = n