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. “
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What is Rank of a Matrix?
Rank of a matrix represents the number of equations that can be formed from the given matrix. Also, the rank of the matrix gives us the number of non-zero rows or columns and the rows or columns that are linearly independent.
Rank Definition
The rank of the matrix is referred to as the number of non-zero rows or columns in the matrix.
The rank of a matrix can also be defined as the number of linearly independent rows in the matrix. The rank of the matrix is denoted as R(A) where A is the matrix.
Properties of Rank of a Matrix
The properties of rank of a matrix are listed below.
- Rank of matrix A = Order of non-singular matrix
- Rank of matrix A = Number of non-zero rows/columns in Echelon form of matrix
- Rank of matrix A < Order of singular matrix A
- Rank of identity matrix = Order of identity matrix
- Rank of null matrix is zero.
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 =
Solution:
First find the determinant of the matrix.
|X| =
⇒ |X| = 1 [(5 × 8) – (2 × 2)] – 6[(4 × 1) – (2× 0)] + 3[(2× 1) – (5× 0)]
⇒ |X| = [40 – 4] – 6 × 4 + 3 × 2
⇒ |X| = 36 – 24 + 6 = 18
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 =
Solution:
First, find the Echelon form of matrix Y by using row operation.
Y =
R2 → R2 – R1
Y =
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.,
- The order r of the identity matrix gives the required rank of the matrix.
Example: Find the rank of matrix B =
Solution:
First, we convert the above matrix into its normal form using elementary row and column operation.
B =
R1 → R1/2
B =
R2→ R2 – R1 and R3 → R3 – 4R1
B =
R2 → R2 / 4.5, R3 → R3 / (-4)
B =
R3 → R3 – R2
B =
C2 → C2 – 1.5C1, C3 → C3 – 2.5C1
B =
C3 → C3 – 3C2
B =
The above matrix is normal form of matrix B i.e., B =
So, the rank of matrix B = R(B) = 2
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Examples on Methods to Find Rank of a Matrix
Example 1: Find the rank of the matrix A =
Solution:
First, find the determinant of the give 2 × 2 matrix.
|A| =
|A| = (4 × 2) – (0 × 5)
|A| = 8 – 0
|A| = 8
Since, determinant of A is not equal to zero.
Rank of A = Order of matrix A
R(A) = 2
Example 2: Find the rank of the matrix B =
Solution:
First, find the Echelon form of matrix B by using row operation.
B =
R1 → R1 / 3
B =
R2 → R2 – R1
B =
The above matrix represents the Echelon form of matrix B.
Now, find the number of non-zero rows i.e., = 2
Rank of matrix B = R(B) = 2
Example 3: Find the rank of the matrix X =
Solution:
First find the determinant of the matrix.
|X| =
|X| = 1 [(3 × 8) – (12 × 2)] – 6[(8 × 1) – (2× 0)] + 4[(12× 1) – (3× 0)]
|X| = [24 – 24] – 6× 8 + 4×12
|X| = 0 – 48 + 48 = 0
Since, |X| = 0 then we find at least one non-zero 2 × 2 minor.
Minor of A =
|Minor of A| = 3 – 6 = -3 ≠ 0
Therefore, rank of matrix X = 2 (order of minor)
Example 4: Find the rank of the matrix D =
Solution:
First convert matrix D into its Echelon form using elementary row operations.
D =
R1 → R1 /2
D =
R2 → R2 – R1
D =
R1 → R1 + R2, and R3 →R3 + R2
D =
R2 → R2 / (-2), R3 → R3/ (-1)
D =
The above matrix is Echelon form of matrix D.
Now, find the number of non-zero rows i.e., = 3
Rank of matrix D = 3
Example 5: Find the rank of the matrix G =
Solution:
First, find the normal form of matrix G using the elementary row and column operations.
G =
R2 → R2 – 2R1 and R3→ R3 + R1
G =
R2 → R2 / (-3)
G =
R3 → R3 + 4R2 and R4→ R4 – 4R2
G =
R4 → R4 + R3
G =
R3 → (3/2)R3 and R4 → R4 /6
G =
C2 → C2 – 2C1 and C3 → C3 – C1
G =
C3 → C3 – (2/3)C2 and C4 → C4 – (1/3)C2
G =
C3 ↔ C4
G =
R3 → R3 / 2 and R4 → R4 – R3
G =
The above matrix is the normal form of matrix G i.e.,
So, the rank of matrix G = R(G) = 3
Practice Problems on Methods to Find Rank of a Matrix
Problem 1: Find the rank of the matrix A =
Problem 2: Find the rank of the matrix B =
Problem 3: Find the rank of the matrix X =
Problem 4: Find the rank of the matrix D =
Problem 5: Find the rank of matrix F =
FAQs on Methods to Find Rank of a Matrix
What is Rank of a Matrix?
The rank of the matrix represents the number of non-zero rows and columns in the matrix.
What is the Rank of 3×3 Matrix?
If the determinant of 3×3 matrix is not zero than the rank of 3×3 matrix is 3 else, we have to find the rank using minors or Echelon form or normal form.
What is the Rank of Null Matrix?
The rank of null matrix is zero.
Can Rank of Matrix be More Than Number of Rows and Columns?
No, the rank of a matrix cannot be more than number of rows and columns.
How to Find Rank of a Matrix?
To find the rank of the matrix we can use one of the following ways:
- By Finding the minors of the matrix
- By using Echelon form