Rank of Matrix Calculator to Find Linear Indepen...
Calculate the rank of a matrix using row reduction (Gaussian elimination). Understand what rank reveals about linear independence, solution existence, and null sp...
What is the Rank of a Matrix?
The rank of a matrix is the number of linearly independent rows (or columns). It equals the number of non-zero rows in row echelon form and determines whether a system of linear equations has no solution, a unique solution, or infinitely many solutions.
📊 Matrix Rank Calculator
Free calculator for instant results.
📐 Formula
rank(A) = dim(col A) = dim(row A)
Found via Gaussian elimination to row echelon form. Rank-nullity theorem: rank(A) + nullity(A) = n (number of columns).
📝 Worked Example
A = [[1,2,3],[2,4,6],[1,1,1]]:
Row 2 = 2×Row 1 (dependent)
After elimination: 2 non-zero rows
rank(A) = 2
Nullity = 3−2 = 1 (one free variable)
📝 How to Use
❓ FAQ
What does full rank mean?
A matrix is full rank if rank = min(m,n). A square full-rank matrix is invertible (non-singular). If rank < min(m,n), the matrix is rank-deficient.
How does rank determine solutions to Ax=b?
Unique solution: rank(A)=rank([A|b])=n. Infinite solutions: rank(A)=rank([A|b])
Veer Kumavat
Founder & AuthorVeer is a 14-year-old student from Nashik, Maharashtra, who built SciFi Calculators to help students worldwide master STEM subjects. He is passionate about making complex science and math problems accessible through intuitive digital tools.