How To Calculate Rank of a Matrix?

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

• Minor Method
• Using Echelon Form
• Using Normal Form

Let’s discuss these methods in detail.

Minor Method

Pre-Requisite: Minors of Matrix

In order to find the rank of a matrix using minor method, following steps are followed:

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

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

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

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

• 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

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. Following steps are followed to calculate the rank of a matrix using 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.

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

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

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

• Step 1: Convert A to echelon form

Apply R₂ = R₂ – 4R₁

Apply R₃ = R₃ – 7R₁

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

Apply R₃ = R₃ – 2R₂

[Tex]A = \begin{bmatrix} 1 & 2 & 3\\ 0 & -3 & -6 \\ 0 & 0 & 0 \end{bmatrix} [/Tex]

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 [Tex] \begin{bmatrix} I_r & 0\\ 0 & 0\\ \end{bmatrix} [/Tex]. Here I_r represents the identity matrix of order r. If a matrix can be converted to its normal form, then rank of the matrix is said to be r.

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

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

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

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

[Tex]A = \begin{bmatrix} 1 & 2 & 1 & 2\\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{bmatrix} [/Tex]

Apply R₁ = R₁ – 2R₂ and R₄ = R₄ – R₂

[Tex]A = \begin{bmatrix} 1 & 0 & -1 & 2\\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} [/Tex]

Apply R₁ = R₁ + R₃ and R₂ = R₂ – R₃

[Tex]A = \begin{bmatrix} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} [/Tex]

Apply C₄ → C₄ – 2C₁

[Tex]A = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} [/Tex]

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

Thus, ρ(A) = 3

Rank of a Matrix is defined as the dimension of the vector space formed by its columns. Rank of a Matrix is a very important concept in the field of Linear Algebra, as it helps us to know if we can find a solution to the system of equations or not. Rank of a matrix also helps us know the dimensionality of its vector space.

This article explores, the concept of the Rank of a Matrix in detail including its definition, how to calculate the rank of the matrix as well as a nullity and its relation with rank. We will also learn how to solve some problems based on the rank of a matrix. So, let’s start with the definition of the rank of the matrix first.