📊 Matrix Rank Calculator
Find the rank of a matrix
How to Use This Calculator
Set Matrix Dimensions
Choose the number of rows and columns (supports 1×1 to 5×5).
Enter Matrix Elements
Input all elements of your matrix.
Calculate
Click to compute the rank using Gaussian elimination.
Review Results
See the rank, RREF, and pivot columns. Rank ≤ min(rows, cols).
Formula
Rank(A) = Number of Pivot Columns in RREF
Maximum number of linearly independent rows/columns
Definition:
The rank of a matrix A is the dimension of the column space (or row space). It equals the number of linearly independent rows or columns.
Computation Method:
- Perform Gaussian elimination to RREF
- Count the number of pivot columns
- This equals the rank
Important Facts:
- Rank(A) = Rank(Aáµ€)
- Rank(A) ≤ min(rows, columns)
- Rank(AB) ≤ min(Rank(A), Rank(B))
- If A is n×n, Rank(A) = n ⟺ A is invertible
- Rank(A) + dim(Null(A)) = number of columns
About Matrix Rank Calculator
The Matrix Rank Calculator determines the rank of a matrix, which is the maximum number of linearly independent rows or columns. Rank is fundamental in linear algebra for characterizing matrix properties and solving systems.
When to Use This Calculator
- Linear Systems: Determine solution structure
- Invertibility: Check if matrix is invertible (full rank)
- Dimensionality: Find dimension of column/row space
- Data Analysis: Identify linearly dependent variables
- Optimization: Understand constraint independence
Why Use Our Calculator?
- ✅ Gaussian Elimination: Uses RREF method
- ✅ RREF Display: Shows reduced row echelon form
- ✅ Pivot Columns: Identifies linearly independent columns
- ✅ Full Rank Check: Indicates if matrix is full rank
- ✅ Educational: Helps understand rank concept
- ✅ Free: No registration required
Key Concepts
- Rank: Dimension of column/row space
- Full Rank: Rank = min(rows, cols)
- Rank Deficient: Rank < min(rows, cols)
- Pivot: Leading nonzero in a row during elimination
- Nullity: Dimension of null space = cols - rank
Applications
System Solvability: Ax = b has solution iff Rank([A|b]) = Rank(A).
Linear Independence: Rank(A) = number of vectors iff they're independent.
Frequently Asked Questions
What is the rank of a matrix?
The rank is the maximum number of linearly independent rows (or columns). It equals the dimension of the row space, column space, and the number of pivot columns in RREF.
Can rank exceed the number of rows or columns?
No! Rank ≤ min(rows, columns). At most, you can have min(rows, cols) independent vectors in an m×n matrix.
What does full rank mean?
Full rank means Rank(A) = min(rows, columns). For square matrices, full rank = n means the matrix is invertible. For rectangular matrices, it means either rows or columns are all independent.
How is rank related to nullity?
Rank-Nullity Theorem: Rank(A) + Nullity(A) = number of columns. Nullity is the dimension of the null space (solutions to Ax = 0).
Can two different matrices have the same rank?
Yes! Many different matrices can have the same rank. Rank only depends on the number of linearly independent rows/columns, not their specific values.