Solved Examples on Rank of 3×3 Matrix
Example 1: Find the rank of 3×3 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) × [-16 -18] – 1 × [16 – 12] + 1 × [12 + 8]
|A| = (-1) × [-34] – 1 × 4 + 1 × 20
|A| = 34 -4 +20
|A| = 50
Since, |A| ≠ 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 ← R1 /(-2)
B = [Tex]\begin{bmatrix} -1&2&3\\ 1&7&-3\\ 0&1& 0 \end{bmatrix}[/Tex]
R2 ← R2 – R1
B = [Tex]\begin{bmatrix} -1&2&3\\ 0&9&0\\ 0&1& 0 \end{bmatrix}[/Tex]
R2 ← R2 / 9
B = [Tex]\begin{bmatrix} -1&2&3\\ 0&1&0\\ 0&1& 0 \end{bmatrix}[/Tex]
R3 ← R3 – 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 3×3 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 ← R1 /2
X = [Tex]\begin{bmatrix} 2&-1/2&3/2\\ 1&-2&1\\ -5&7&-6 \end {bmatrix}[/Tex]
R2 ← R2 – R1, R3 ← 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 ← 2C2, C3 ← 2C3
X = [Tex]\begin{bmatrix} 1&-1&3\\ 0&-3&-1\\ 0&9&3 \end {bmatrix}[/Tex]
R2 ← R2 / (-3), R3 ← R3/3
X = [Tex]\begin{bmatrix} 1&-1&3\\ 0&1&1/3\\ 0&3&1 \end {bmatrix}[/Tex]
R1 ← R1 + R2, R3 ← R3 – 3R2
X = [Tex]\begin{bmatrix} 1&0&10/3\\ 0&1&1/3\\ 0&0&0 \end {bmatrix}[/Tex]
C3 ← C3 – (C2 / 3)
X = [Tex]\begin{bmatrix} 1&0&10/3\\ 0&1&0\\ 0&0&0 \end {bmatrix}[/Tex]
C3 ← C3 – (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 3×3 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 3×3 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 3×3 Matrix
- Solved Examples on Rank of 3×3 Matrix