Matrix Rank

Overview

The rank of a matrix is the dimension of its column space (or row space). It measures the number of linearly independent rows or columns.

Definition

For matrix $A$:

$\text{rank}(A) = \dim(\text{Col}(A)) = \dim(\text{Row}(A))$

This equals:

• Number of pivot positions in echelon form
• Number of linearly independent columns
• Number of linearly independent rows

Finding Rank

Method 1: Row Reduction

1. Row reduce $A$ to echelon form
2. Count the number of pivot positions (leading 1s)

Example

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \xrightarrow{\text{RREF}} \begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$

$\text{rank}(A) = 2$ (two pivots)

Key Properties

Property	Formula
Maximum	$\text{rank}(A) \leq \min(m, n)$ for $m \times n$ matrix
Transpose	$\text{rank}(A) = \text{rank}(A^T)$
Sum	$\text{rank}(A + B) \leq \text{rank}(A) + \text{rank}(B)$
Product	$\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))$
Inverse	If $A$ is invertible, $\text{rank}(A) = n$

Full Rank

Full Column Rank

$\text{rank}(A) = n$ (number of columns)

• Columns are linearly independent
• $A\mathbf{x} = \mathbf{0}$ has only the trivial solution
• $A\mathbf{x} = \mathbf{b}$ has at most one solution

Full Row Rank

$\text{rank}(A) = m$ (number of rows)

• Rows are linearly independent
• $A\mathbf{x} = \mathbf{b}$ has at least one solution for every $\mathbf{b}$

Full Rank (Square Matrix)

$\text{rank}(A) = m = n$

• $A$ is invertible
• $A\mathbf{x} = \mathbf{b}$ has a unique solution for every $\mathbf{b}$

Rank-Nullity Theorem

For an $m \times n$ matrix $A$:

$\text{rank}(A) + \text{nullity}(A) = n$

where $\text{nullity}(A) = \dim(\text{Null}(A))$ = number of free variables.

Example

For a $3 \times 5$ matrix with rank 2:

$\text{nullity} = 5 - 2 = 3$

The null space has dimension 3.

Rank and Systems of Equations

For the system $A\mathbf{x} = \mathbf{b}$ ($A$ is $m \times n$):

Condition	Solutions
$\text{rank}(A) < \text{rank}([A \mid \mathbf{b}])$	No solution
$\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) = n$	Unique solution
$\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) < n$	Infinitely many solutions

Examples

Example 1: Find Rank

$A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 3 \\ 3 & 6 & 4 \end{bmatrix}$

Row reduce:

$\begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 3 \\ 3 & 6 & 4 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$

$\text{rank}(A) = 2$

Example 2: Apply Rank-Nullity

Matrix $A$ is $4 \times 6$ with rank 3.

$\text{nullity}(A) = 6 - 3 = 3$

$A\mathbf{x} = \mathbf{0}$ has a 3-dimensional solution space.

Example 3: Check Solvability

For $A\mathbf{x} = \mathbf{b}$ with $A$ ($3 \times 3$) having rank 2:

$\left[\begin{array}{ccc|c} 1 & 2 & 3 & b_1 \\ 2 & 4 & 6 & b_2 \\ 1 & 2 & 3 & b_3 \end{array}\right] \to \left[\begin{array}{ccc|c} 1 & 2 & 3 & b_1 \\ 0 & 0 & 0 & b_2 - 2b_1 \\ 0 & 0 & 0 & b_3 - b_1 \end{array}\right]$

For solution: $b_2 = 2b_1$ and $b_3 = b_1$

Otherwise: no solution

Rank and Determinant

For an $n \times n$ matrix $A$:

• $\text{rank}(A) = n \Leftrightarrow \det(A) \neq 0$
• $\text{rank}(A) < n \Leftrightarrow \det(A) = 0$