Rank and Nullity

Rank and Nullity are two essential concepts related to matrices in Linear Algebra. The nullity of a matrix is determined by the difference between the order and rank of the matrix. The rank of a matrix is the number of linearly independent row or column vectors of a matrix. If n is the order of the square matrix A, then the nullity of A is given by n – r. Thus, the rank of a matrix is the number of linearly independent or non-zero vectors of a matrix, whereas nullity is the number of zero vectors of a matrix.

The rank of matrix A is denoted as ρ(A), and the nullity is denoted as N(A). Evidently, if the rank of the matrix is equal to the order of the matrix, then the nullity of the matrix is zero.

Rank and Nullity Theorem for Matrix

The rank and nullity theorem for matrices is one of the important theorems in linear algebra and a requirement to derive many more results. Before discussing the theorem, we must know the concept of null spaces.

Nullspace

Let A be a real matrix of order m × n, the set of the solutions associated with the system of homogeneous equation AX = 0 is said to be the null space of A.

Nullspace of A = { x ∈ Rⁿ | Ax = 0}. Then the nullity of A will be the dimension of the Nullspace of A.

If the rank of A is r, there are r leading variables in row-reduced echelon form of A and n – r free variables, which are solutions of the homogeneous system of equation AX = 0. Thus, n – r is the dimension of the null space of A. This fact motivates the rank and nullity theorem for matrices.

Row-reduced Echelon Form of A

Rank and Nullity Theorem

Proof: We already have a result, “Let A be a matrix of order m × n, then the rank of A is equal to the number of leading columns of row-reduced echelon form of A.”

Let r be the rank of A, and then we have two cases as follows:

Case I: If A is a non-singular matrix, then r = n as the row-reduced echelon form will have no zero rows. Then, x = 0 will be the trivial solution of AX = 0; that is, the nullity of A will be zero.

Hence, rank of A + nullity of A = n + 0 = n = Number of columns in A

Case II: If A is singular, then rank of will less than the order of A, that is, r < n. Therefore, there are r non-zero or r leading columns of the row-reduced echelon form. Consequently, there will n – r zero rows, which contributes to the solution of AX = 0, which means the nullity of A is n – r.

Hence, rank of A + nullity of A = r + n – r = n = Number of columns in A.

In both cases, we get the same result which proves our claim.

Important Facts on Rank and Nullity

• The rank of an invertible matrix is equal to the order of the matrix, and its nullity is equal to zero.
• Rank is the number of leading column or non-zero row vectors of row-reduced echelon form of the given matrix, and the number of zero columns is the nullity.
• The nullity of a matrix is the dimension of the null space of A, also called the kernel of A.
• If A is an invertible matrix, then null space (A) = {0}.
• The rank of a matrix is the number of non-zero eigenvalues of the matrix, and the number of zero eigenvalues determines the nullity of the matrix.

Related Articles

• Determinants and Matrices
• Determinant of a Matrix
• Matrix Multiplication
• Matrix Operations

Solved Examples on Rank and Nullity

Example 1:

Verify the rank and nullity theorem for the matrix

Solution:

Given

Let us reduce this in row reduced echelon form

Applying R₂ → R₂ + (-3)R₁ and R₃ → R₃ + R₁

Applying R₃ → R₃ + R₂

Applying C₂ → C₂ + C₁, C₃ → C₃ + (-2)C₁ and C₄ → C₄ + (-3)C₁

Applying C₃ → C₃ + 7C₂ and C₄ → C₄ + 7C₂

Clearly, rank(A) = 2 and nullity(A) = 2

Therefore, rank(A) + nullity(2) = 2 + 2 = 4 = Number of columns.

Example 2:

Find the nullity of the matrix

Solution:

Given matrix

Applying elementary operations, R₂₁(-3) and R₃₁(-1)

Applying elementary operations, R₃(-½)

Applying elementary operations, C₂₄, we get

Applying elementary operations, C₂₁(-3), C₃₁(-4) and C₄₁(-3) we get

Thus, Rank of matrix A = 2 and Nullity = Number of columns – rank = 4 – 2 = 2.