Matrix Rank
The rank of a matrix is the dimension of the vector space spanned by its columns (column rank), which always equals the dimension spanned by its rows (row rank), giving a single well-defined measure of the "information content" or degrees of freedom in the matrix. A matrix A of size m×n has rank at most min(m, n); if the rank equals min(m, n) the matrix is called full rank. The rank determines whether a linear system Ax = b has a unique solution (full rank square matrix), infinitely many solutions, or no solution, making it central to the theory of linear equations, least-squares fitting, and dimensionality reduction.
Key Formula
rank(A) + nullity(A) = n (number of columns)
LaTeX: \text{rank}(A) + \text{nullity}(A) = n
| Symbol | Meaning | Unit |
|---|---|---|
| \text{rank}(A) | Dimension of column space of A | — |
| \text{nullity}(A) | Dimension of null space (kernel) of A | — |
| n | Number of columns of A | — |
Worked Example
Problem
Find the rank of A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Solution
Step 1: Row reduce A. Step 2: R2 ← R2 − 4R1: [[1, 2, 3], [0, −3, −6], [7, 8, 9]]. Step 3: R3 ← R3 − 7R1: [[1, 2, 3], [0, −3, −6], [0, −6, −12]]. Step 4: R3 ← R3 − 2R2: [[1, 2, 3], [0, −3, −6], [0, 0, 0]]. Step 5: Count non-zero rows in row echelon form: 2.
Answer
rank(A) = 2. The matrix is rank-deficient (rank < 3), confirming the rows are linearly dependent.
Rank and Solutions to Linear Systems Ax = b
| Condition | rank(A) | rank([A|b]) | Solution Type |
|---|---|---|---|
| Unique solution | n (full rank) | n | Exactly one solution |
| Infinitely many solutions | r < n | r (same as rank A) | Free variables exist |
| No solution | r | r + 1 (> rank A) | Inconsistent system |
| Full rank square (m=n) | n | n | Unique solution |
| Overdetermined consistent | r ≤ n | r | Least-squares best fit |
Interactive Tools
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Related Terms
Matrix (Linear Algebra)
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns, where an m×n matrix has m rows and n columns. Matrices represent linear transformations between vector spaces, encode systems of linear equations, and serve as the primary computational tool in linear algebra. They are indispensable across engineering, physics, computer science, statistics, and machine learning, where they model data, rotations, reflections, and the weights of neural networks.
Linear Independence
A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others, equivalently if the only solution to the equation c₁v₁ + c₂v₂ + … + cₙvₙ = 0 is the trivial solution where all scalars cᵢ equal zero. If at least one non-zero scalar solution exists, the set is called linearly dependent. Linear independence is the key criterion for determining whether a set of vectors forms a basis, and it underlies the concepts of dimension, rank, and solutions to linear systems.
Determinant
The determinant is a scalar value computed from a square matrix that encodes whether the linear transformation represented by the matrix is invertible, scaling the volume by a factor equal to the absolute value of the determinant and changing orientation if the determinant is negative. For a 2×2 matrix, det(A) = ad − bc; for larger matrices, it is computed recursively via cofactor expansion or row reduction. The determinant is zero if and only if the matrix is singular (non-invertible), making it a critical tool in solving linear systems, computing eigenvalues, and computing cross products in geometry.
From French "rang" and Latin "rangus" (row, line, order). In linear algebra, "rank" as a measure of the number of independent rows/columns was formalised in the 19th century, with significant contributions from Frobenius and Sylvester.