Matrix Rank Calculator
Calculate the rank of a matrix using Gaussian elimination with step-by-step solutions
Matrix Input
Matrix Elements
Enter matrix elements. Use decimal numbers (e.g., 0.5, -2.3)
Quick Examples
Matrix Rank Results
Matrix Properties
- Full Row Rank:No
- Full Column Rank:No
- Invertible:No
Rank Formula
Rank: Number of linearly independent rows/columns
Nullity: n - rank(A) = 3 - 2 = 1
Rank-Nullity Theorem: rank + nullity = 3
Step-by-Step Solution
Row Echelon Form
This is the row echelon form obtained through Gaussian elimination
Matrix Rank Guide
Set Dimensions
Choose matrix size (up to 5×5)
Enter Elements
Fill in matrix values
Get Results
Rank calculated automatically
Key Concepts
Rank: Number of linearly independent rows or columns
Full Rank: rank(A) = min(m,n)
Nullity: n - rank(A)
Gaussian Elimination: Method to find rank
Understanding Matrix Rank
What is Matrix Rank?
The rank of a matrix is the maximum number of linearly independent rows (or equivalently, columns) in the matrix. It represents the dimension of the vector space spanned by the matrix's rows or columns.
How to Calculate Rank
- •Use Gaussian elimination to transform matrix to row echelon form
- •Count the number of non-zero rows
- •This count equals the matrix rank
Important Properties
- Rank-Nullity Theorem: rank(A) + nullity(A) = n
- Maximum Rank: rank(A) ≤ min(m, n)
- Full Rank: rank(A) = min(m, n)
- Invertible: Square matrix A is invertible ↔ rank(A) = n
Applications
- •Solving systems of linear equations
- •Determining matrix invertibility
- •Linear independence verification