Solved Examples on Rank of 3Γ3 Matrix
Example 1: Find the rank of 3Γ3 Matrix A = [Tex]\begin{bmatrix} -1&1&1\\ 2&-2&3\\ 4&6& 8 \end{bmatrix}[/Tex] using minor method.
Solution:
First find the determinant of A
|A| = [Tex]\begin{vmatrix} -1&1&1\\ 2&-2&3\\ 4&6& 8 \end{vmatrix}[/Tex]
|A| = (-1) Γ [-16 -18] β 1 Γ [16 β 12] + 1 Γ [12 + 8]
|A| = (-1) Γ [-34] β 1 Γ 4 + 1 Γ 20
|A| = 34 -4 +20
|A| = 50
Since, |A| β 0
Rank of matrix = 3
Example 2: Calculate the rank of matrix B = [Tex]\begin{bmatrix} -2&4&6\\ 1&7&-3\\ 0&1& 0 \end{bmatrix}[/Tex] using Echelon form.
Solution:
B = [Tex]\begin{bmatrix} -2&4&6\\ 1&7&-3\\ 0&1& 0 \end{bmatrix}[/Tex]
R1 β R1 /(-2)
B = [Tex]\begin{bmatrix} -1&2&3\\ 1&7&-3\\ 0&1& 0 \end{bmatrix}[/Tex]
R2 β R2 β R1
B = [Tex]\begin{bmatrix} -1&2&3\\ 0&9&0\\ 0&1& 0 \end{bmatrix}[/Tex]
R2 β R2 / 9
B = [Tex]\begin{bmatrix} -1&2&3\\ 0&1&0\\ 0&1& 0 \end{bmatrix}[/Tex]
R3 β R3 β R2
B = [Tex]\begin{bmatrix} -1&2&3\\ 0&1&0\\ 0&0& 0 \end{bmatrix}[/Tex]
In the above matrix number of non-zero rows = 2
So, the rank of matrix B = 2
Example 3: Find the rank of 3Γ3 matrix X = [Tex]\begin{bmatrix} 2&-1&3\\ 1&-2&1\\ -5&7&-6 \end {bmatrix}[/Tex] using normal form.
Solution:
X = [Tex]\begin{bmatrix} 2&-1&3\\ 1&-2&1\\ -5&7&-6 \end {bmatrix}[/Tex]
R1 β R1 /2
X = [Tex]\begin{bmatrix} 2&-1/2&3/2\\ 1&-2&1\\ -5&7&-6 \end {bmatrix}[/Tex]
R2 β R2 β R1, R3 β R3 + 5R1
X = [Tex]\begin{bmatrix} 2&-1/2&3/2\\ 0&-3/2&-1/2\\ 0&9/2&3/2 \end {bmatrix}[/Tex]
C2 β 2C2, C3 β 2C3
X = [Tex]\begin{bmatrix} 1&-1&3\\ 0&-3&-1\\ 0&9&3 \end {bmatrix}[/Tex]
R2 β R2 / (-3), R3 β R3/3
X = [Tex]\begin{bmatrix} 1&-1&3\\ 0&1&1/3\\ 0&3&1 \end {bmatrix}[/Tex]
R1 β R1 + R2, R3 β R3 β 3R2
X = [Tex]\begin{bmatrix} 1&0&10/3\\ 0&1&1/3\\ 0&0&0 \end {bmatrix}[/Tex]
C3 β C3 β (C2 / 3)
X = [Tex]\begin{bmatrix} 1&0&10/3\\ 0&1&0\\ 0&0&0 \end {bmatrix}[/Tex]
C3 β C3 β (10 / 3) C1
X = [Tex]\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&0 \end {bmatrix}[/Tex]
So, the normal form of matrix X = [Tex]\begin{bmatrix} I_2&0\\ 0&0\\ \end {bmatrix}[/Tex]
Rank of matrix X = 3
How to Find Rank of a 3Γ3 Matrix
Rank of a matrix is equal to the number of linear independent rows or columns in it. The rank of the matrix is always less than or equal to the order of the matrix.
In this article we will explore how to find rank of 3Γ3 matrix in detail along with the basics of the rank of a matrix.
Table of Content
- What is Rank of a Matrix?
- How to Find Rank of a 3Γ3 Matrix
- Solved Examples on Rank of 3Γ3 Matrix
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