Matrix Rank Calculator
Calculate the rank and nullity of any matrix. Enter your matrix below and get instant results with detailed step-by-step row reduction process.
Matrix A
Row Echelon Form
How to Calculate Matrix Rank
Row Reduction
Use elementary row operations to transform matrix to row echelon form and count non-zero rows.
Linear Independence
Rank equals the maximum number of linearly independent rows or columns in the matrix.
Nullity Theorem
For an m×n matrix: Rank + Nullity = n (number of columns). This is the rank-nullity theorem.
Full Rank
Matrix has full rank when rank equals min(m,n). Square matrices with full rank are invertible.
Mathematical Theory & Applications
What is Matrix Rank?
The rank of a matrix is the dimension of the vector space spanned by its rows (or columns). It represents the maximum number of linearly independent rows or columns in the matrix. The rank provides crucial information about the matrix's properties and the linear system it represents.
Definition: rank(A) = dim(row space) = dim(column space)
Rank-Nullity Theorem: rank(A) + nullity(A) = n
Where n is the number of columns in matrix A
Historical Development
The concept of matrix rank was developed in the 19th century as part of the systematic study of linear algebra. Carl Friedrich Gauss laid the groundwork with his elimination method, while Georg Frobenius formalized many of the rank theorems we use today.
The rank-nullity theorem, one of the fundamental results in linear algebra, was established through the work of multiple mathematicians and provides a deep connection between the dimensions of different vector spaces associated with a linear transformation.
Properties of Matrix Rank
Rank Bounds
0 ≤ rank(A) ≤ min(m,n)
Transpose Property
rank(A) = rank(AT)
Product Rule
rank(AB) ≤ min(rank(A), rank(B))
Sum Rule
rank(A + B) ≤ rank(A) + rank(B)
Real-World Applications
Data Analysis
Determining effective dimensionality in datasets and principal component analysis
System Solvability
Analyzing whether linear systems have unique, infinite, or no solutions
Image Processing
Low-rank approximations for image compression and noise reduction
Machine Learning
Feature selection, dimensionality reduction, and model complexity analysis
Frequently Asked Questions
Rank is the number of linearly independent rows/columns, while nullity is the dimension of the null space (solutions to Ax = 0). They're related by the rank-nullity theorem: rank + nullity = number of columns.
A matrix has full rank when its rank equals min(rows, columns). For square matrices, full rank means the matrix is invertible and its determinant is non-zero. For rectangular matrices, it means maximum possible rank.
Rank determines solution existence and uniqueness. If rank(A) = rank([A|b]) = n, there's a unique solution. If rank(A) = rank([A|b]) < n, infinite solutions exist. If rank(A) ≠ rank([A|b]), no solution exists.
No! A fundamental theorem states that row rank always equals column rank. This common value is simply called "the rank" of the matrix. This equality is not obvious but is a deep result in linear algebra.
This means some rows are linearly dependent - they can be expressed as combinations of other rows. The "missing" rank represents redundant information. In applications, this often indicates constraints or relationships in the data.
Rank helps identify effective data dimensionality, detect multicollinearity in features, perform dimensionality reduction (PCA), and create low-rank approximations for compression and noise reduction. It's fundamental to understanding data structure.
Common methods include Gaussian elimination (row reduction), LU decomposition, QR decomposition, and SVD (Singular Value Decomposition). SVD is most numerically stable for rank determination, especially with floating-point precision issues.