Methods to Find Rank of a Matrix

The two methods to find the rank of the matrix are:

• Rank of Matrix by Finding Minors
• Rank of Matrix by Finding Echelon Form
• Rank of Matrix by Finding Normal Form

Let’s discuss each method in detail.

Rank of Matrix by Finding Minors

Steps to find rank of matrix by finding minors are listed below.

• First, find the determinant of the matrix.
• If the determinant of matrix ≠ 0 then, rank of matrix = order of matrix.
• If the determinant of matrix = 0 then, the rank of matrix = maximum order of one of the minors which is non-zero.

Example: Find the rank of matrix M = [Tex]\begin {bmatrix} 1 & 6 & 3\\ 1 & 5 & 2\\ 0 & 2 & 4 \end {bmatrix}[/Tex] using minor method.

Solution:

First find the determinant of the matrix.

|X| = [Tex]\begin {vmatrix} 1 & 6 & 3\\ 1 & 5 & 2\\ 0 & 2 & 4 \end {vmatrix}[/Tex]

⇒ |X| = 1 [(5 × 4) – (2 × 2)] – 6[(4 × 1) – (2× 0)] + 3[(2× 1) – (5× 0)]

⇒ |X| = [20 – 4] – 6 × 4 + 3 × 2

⇒ |X| = 16 – 24 + 6 = -2

Since |X| ≠ 0 so,

Rank of matrix = Order of matrix

Rank of matrix M = 3

Rank of Matrix by Finding Echelon Form

Steps to find rank of matrix by finding Echelon form are listed below.

• First convert the matrix into its Echelon form (i.e., upper triangular matrix or lower triangular matrix) using elementary row operations.
• Then, after converting the matrix into its Echelon form find the number of non-zero rows.
• The rank of the matrix is given by the number of non-zero rows present in Echelon form of matrix.

Example: Find the rank of matrix Y = [Tex]\begin {bmatrix} 1 & -1\\ 1 & 0 \end {bmatrix}[/Tex] using Echelon form.

Solution:

First, find the Echelon form of matrix Y by using row operation.

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

R₂ → R₂ – R₁

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

The above matrix represents the Echelon form of matrix Y.

Now, find the number of non-zero rows i.e., = 2.

So, the rank of matrix Y = R(Y) = 2

Rank of Matrix by Finding Normal Form

Steps to find rank of matrix by finding normal form are listed below.

• First try to convert the given matrix into its normal form using the elementary row and column operation.
• If the matrix can be converted into its normal form i.e., [Tex]\begin{bmatrix} I_r &0\\ 0 & 0 \end{bmatrix}[/Tex]
• The order r of the identity matrix gives the required rank of the matrix.

Example: Find the rank of matrix B = [Tex]\begin {bmatrix} 2 & 3&5\\ 1 & 6 & 16\\ 4 & 2&-2 \end {bmatrix}[/Tex] using normal form method.

Solution:

First, we convert the above matrix into its normal form using elementary row and column operation.

B = [Tex]\begin {bmatrix} 2 & 3&5\\ 1 & 6 & 16\\ 4 & 2&-2 \end {bmatrix}[/Tex]

R₁ → R₁/2

B = [Tex]\begin {bmatrix} 1 & 1.5& 2.5\\ 1 & 6 & 16\\ 4 & 2&-2 \end {bmatrix}[/Tex]

R₂→ R₂ – R₁ and R₃ → R₃ – 4R₁

B = [Tex]\begin {bmatrix} 1 & 1.5& 2.5\\ 0& 4.5 & 13.5\\ 0&-4&-12 \end {bmatrix}[/Tex]

R₂ → R₂ / 4.5, R₃ → R₃ / (-4)

B = [Tex]\begin {bmatrix} 1 & 1.5& 2.5\\ 0& 1 & 3\\ 0&1&3 \end {bmatrix}[/Tex]

R₃ → R₃ – R₂

B = [Tex]\begin {bmatrix} 1 & 1.5& 2.5\\ 0& 1 & 3\\ 0&0&0 \end {bmatrix}[/Tex]

C₂ → C₂ – 1.5C₁, C₃ → C₃ – 2.5C₁

B = [Tex]\begin {bmatrix} 1 & 0 & 0\\ 0& 1 & 3\\ 0&0&0 \end {bmatrix}[/Tex]

C₃ → C₃ – 3C₂

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

The above matrix is normal form of matrix B i.e., B = [Tex]\begin {bmatrix} I_2 & 0 \\ 0& 0\\ \end {bmatrix}[/Tex]

So, the rank of matrix B = R(B) = 2

Read More,

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

To find the rank of a matrix find the highest order of the non-zero minor within the matrix. Rank of a matrix in the number that represents the number of non-zeros rows or columns in the matrix. If the rank of the matrix is r then the matrix contains at least one minor with order r and the minors with order greater than r is zero. The second method to find the rank of matrix is by converting it into Echelon form.

In this article we will discuss methods to find rank of a matrix in depth along with the rank definition, methods to find rank of a matrix i.e., by minors and by Echelon form. Also, we will discuss the properties of rank and solve some examples including both the methods. Let’s start our learning on the topic “Methods to Find Rank of a Matrix. “