Rank of a Matrix: Definition, Properties, and Formula

The Rank of a Matrix is a fundamental concept in linear algebra that measures the number of linearly independent rows or columns in a matrix. It essentially determines the dimensionality of the vector space formed by the rows or columns of the matrix.

It helps determine:

• If a system of linear equations has solutions.
• The "usefulness" of rows or columns contributes to the matrix's information.

Definition of Rank of a Matrix

The rank of a matrix is denoted using ρ(A) where A is any matrix. Thus the number of rows of a matrix is a limit on the rank of the matrix, which means the rank of the matrix cannot exceed the total number of rows in a matrix.

For example, if a matrix is of the order 3×3 then the maximum rank of a matrix can be 3.

Note: If a matrix has all rows with zero elements, then the rank of a matrix is said to be zero.

Nullity of Matrix

In a given matrix, the number of vectors in the null space is called the nullity of the matrix or it can also be defined as the dimension of the null space of the given matrix.

Total columns in a matrix = Rank + Nullity

Read More about the Rank Nullity Theorem.

Rank of a Matrix Formulas

3 methods can be used to get the rank of any given matrix. These methods are as follows:

• Minor Method
• Using Echelon Form
• Using Normal Form

Using Minor Method

Pre-Requisite: Minors of Matrix

To find the rank of a matrix using the minor method, the following steps are followed:

• Calculate the determinant of the matrix (say A). If det(A) ≠ 0, then the rank of matrix A = order of matrix A.
• If det(A) = 0, then the rank of the matrix is equal to the order of the maximum possible nonzero minor of the matrix.

Let us understand how to find the rank of a matrix using the minor method.

Example: Find the rank of a matrix \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6 \\ 7 & 8 & 7\end{bmatrix} using the minor method.

Given A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6 \\ 7 & 8 & 7\end{bmatrix}

• Step 1: Calculate the determinant of A

det(A) = 1 (35 - 48) - 2 (28 - 42) + 3 (32 - 35)
det(A) = -13 + 28 - 9 = 6

• As det(A) ≠ 0,
  ρ(A) = order of A = 3

Using Echelon Form

The minor method becomes very tedious if the order of the matrix is very large. So in this case, we convert the matrix into Echelon Form. A matrix that is in upper triangular form or lower triangular form is considered to be in Echelon Form. A matrix can be converted to its Echelon Form by using elementary row operations. The following steps are followed to calculate the rank of a matrix using the Echelon form:

• Convert the given matrix into its Echelon Form.
• The number of non-zero rows obtained in the Echelon form of the matrix is the rank of the matrix.

Read more - Triangular Matrix

Let us understand how to find the rank of a matrix using the minor method.

Example: Find the rank of a matrix \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}using the Echelon form method.

Given A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}

• Step 1: Convert A to echelon form

Apply R₂ = R₂ - 4R₁
Apply R₃ = R₃ - 7R₁

A = \begin{bmatrix}1 & 2 & 3\\0 & -3 & -6 \\ 0 & -6 & -12\end{bmatrix}

Apply R₃ = R₃ - 2R₂

A = \begin{bmatrix}1 & 2 & 3\\0 & -3 & -6 \\ 0 & 0 & 0\end{bmatrix}

As matrix A is now in lower triangular form, it is in Echelon Form.

• Step 2: Number of non-zero rows in A = 2.
  Thus ρ(A) = 2

Using Normal Form

A matrix is said to be in normal form if it can be reduced to the form \begin{bmatrix} I_r & 0\\ 0 & 0\\ \end{bmatrix} . Here I_r represents the identity matrix of order r. If a matrix can be converted to its normal form, then the rank of the matrix is said to be r.

Let us understand how to find the rank of a matrix using the minor method.

Example: Find the rank of a matrix \bold{\begin{bmatrix}1 & 2 & 1 & 2\\1 & 3 & 2 & 2 \\ 2 & 4 & 3 & 4 \\3 & 7 & 4 & 6\end{bmatrix}}using the normal form method.

Solution:

Given A = \begin{bmatrix}1 & 2 & 1 & 2\\1 & 3 & 2 & 2 \\ 2 & 4 & 3 & 4 \\3 & 7 & 4 & 6\end{bmatrix}

Apply R₂ = R₂ - R₁ , R₃ = R₃ - 2R₁ and R₄ = R₄ - 3R₁

A = \begin{bmatrix}1 & 2 & 1 & 2\\0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\0 & 1 & 1 & 0\end{bmatrix}

Apply R₁ = R₁ - 2R₂ and R₄ = R₄ - R₂

A = \begin{bmatrix}1 & 0 & -1 & 2\\0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0\end{bmatrix}

Apply R₁ = R₁ + R₃ and R₂ = R₂ - R₃

A = \begin{bmatrix}1 & 0 & 0 & 2\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0\end{bmatrix}

Apply C₄ → C₄ - 2C₁

A = \begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0\end{bmatrix}

Thus A can be written as \begin{bmatrix} I_3 & 0\\ 0 & 0\\ \end{bmatrix} .

Thus, ρ(A) = 3

Properties of Rank of Matrix

The properties of the rank of a matrix are as follows:

• The rank of a matrix is equal to the order of the matrix if it is a non-singular matrix.
• The rank of a matrix is equal to the number of non-zero rows if it is in Echelon Form.
• The rank of a matrix is equal to the order of the identity matrix in it if it is in normal form.
• Rank of matrix < Order of matrix if it is a singular matrix.
• Rank of matrix < minimum {m, n} if it is a rectangular matrix of order m x n.
• The rank of the identity matrix is equal to the order of the identity matrix.
• The rank of a zero matrix or a null matrix is zero.

Read More,

• Matrices
• Types of Matrices
• Transpose of a Matrix
• Inverse of Matrix

Solved Examples Rank of a Matrix

Example 1: Find the rank of a matrix \bold{\begin{bmatrix}-1 & -2 & -3\\-4 & -5 & -6 \\ -7 & -8 & -7\end{bmatrix}}using the minor method.

Solution:

Given A = \begin{bmatrix}-1 & -2 & -3\\-4 & -5 & -6 \\ -7 & -8 & -7\end{bmatrix}

Step 1: Calculate the determinant of A
det(A) = -1 (35 - 48) + 2 (28 - 42) - 3 (32 - 35)
det(A) = 13 - 28 - 9 = -24
As det(A) ≠ 0, ρ(A) = order of A = 3

Example 2. Find the rank of a matrix \bold{\begin{bmatrix}2 & 4 & 6\\8 & 10 & 12 \\ 14 & 16 & 0\end{bmatrix}}using the minor method.

Solution:

Given A = \begin{bmatrix}2 & 4 & 6\\8 & 10 & 12 \\ 14 & 16 & 0\end{bmatrix}

Step 1: Calculate the determinant of A
det(A) = 2(0-192) - 4(0-168) + 6(128-140)
det(A) = -384 + 672 - 72 = 216
As det(A) ≠ 0, ρ(A) = order of A = 3

Example 3. Find the rank of a matrix \bold{\begin{bmatrix}-1 & -2 & -3\\-4 & -5 & -6 \\ -7 & -8 & -9\end{bmatrix}}using the Echelon form method.

Solution:

Given A = \begin{bmatrix}-1 & -2 & -3\\-4 & -5 & -6 \\ -7 & -8 & -9\end{bmatrix}

Step 1: Convert A to echelon form
Apply R₂ = R₂ - 4R₁
Apply R₃ = R₃ - 7R₁

A = \begin{bmatrix}-1 & -2 & -3\\0 & 3 & 6 \\ 0 & 6 & 12\end{bmatrix}

Apply R₃ = R₃ - 2R₂

A = \begin{bmatrix}-1 & -2 & -3\\0 & 3 & 6 \\ 0 & 0 & 0\end{bmatrix}

As matrix A is now in lower triangular form, it is in Echelon Form.

Step 2: Number of non-zero rows in A = 2.
Thus ρ(A) = 2

Example 4. Find the rank of a matrix \bold{\begin{bmatrix}2 & 4 & 6\\8 & 10 & 12 \\ 14 & 16 & 18\end{bmatrix}}using the Echelon form method.

Solution:

Given A = \begin{bmatrix}2 & 4 & 6\\8 & 10 & 12 \\ 14 & 16 & 18\end{bmatrix}

Step 1: Convert A to echelon form
Apply R₂ = R₂ - 4R₁
Apply R₃ = R₃ - 7R₁

A = \begin{bmatrix}2 & 4 & 6\\0 & -6 & -12 \\ 0 & -12 & -24\end{bmatrix}

Apply R₃ = R₃ - 2R₂

A = \begin{bmatrix}2 & 4 & 6\\0 & -6 & -12 \\ 0 & 0 & 0\end{bmatrix}

As matrix A is now in lower triangular form, it is in Echelon Form.

Step 2: Number of non-zero rows in A = 2.
Thus ρ(A) = 2

Example 5. Find the rank of a matrix \bold{\begin{bmatrix}2 & 4 & 2 & 4\\2 & 6 & 4 & 4 \\ 4 & 8 & 6 & 8 \\6 & 14 & 8 & 12\end{bmatrix}}using the normal form method.

Solution:

Given A = \begin{bmatrix}2 & 4 & 2 & 4\\2 & 6 & 4 & 4 \\ 4 & 8 & 6 & 8 \\6 & 14 & 8 & 12\end{bmatrix}

Apply R₂ = R₂ - R₁ , R₃ = R₃ - 2R₁ and R₄ = R₄ - 3R₁

A = \begin{bmatrix}2 & 4 & 2 & 4\\0 & 2 & 2 & 0 \\ 0 & 0 & 2 & 0 \\0 & 2 & 2 & 0\end{bmatrix}

Apply R₁ = R₁ - 2R₂ and R4 = R₄ - R₂

A = \begin{bmatrix}2 & 0 & -2 & 4\\0 & 2 & 2 & 0 \\ 0 & 0 & 2 & 0 \\0 & 0 & 0 & 0\end{bmatrix}

Apply R₁ = R₁ + R₃ and R₂ = R₂ - R₃

A = \begin{bmatrix}2 & 0 & 0 & 4\\0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\0 & 0 & 0 & 0\end{bmatrix}

Apply C₄ → C₄ - 2C₁

A = \begin{bmatrix}2 & 0 & 0 & 0\\0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\0 & 0 & 0 & 0\end{bmatrix}

Apply R₁ = R₁/2, R₂ = R₂/2, R₃ = R₃/2

A = \begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0\end{bmatrix}

Thus A can be written as \begin{bmatrix} I_3 & 0\\ 0 & 0\\ \end{bmatrix}
Thus, ρ(A) = 3

What is the Rank of a Matrix? Formula and Examples

What is the Rank of a Matrix? Formula and Examples

Rank of a Matrix: Definition, Properties, and Formula