Q.46 Value of the determinant mentioned below is
| 1 | 0 | −1 | 0 |
| 4 | 7 | 0 | 2 |
| 1 | 1 | −1 | 1 |
| 2 | 0 | 2 | 1 |
Introduction
Finding the value of the determinant is an important topic in matrix algebra.
Determinants are widely used in competitive exams to test matrix manipulation skills.
In this problem, we calculate the determinant of a 4×4 matrix step by step and analyze all given options.
Given Question (Q.46)
Find the value of the determinant:
| 1 0 -1 0 |
| 4 7 0 2 |
| 1 1 -1 1 |
| 2 0 2 1 |
Options:
- (A) 24
- (B) −30
- (C) −24
- (D) −10
Step-by-Step Solution
We expand the determinant along the first row because it contains zeros.
= 1 × M11 − 0 + (−1) × M13 − 0
Step 1: Calculation of M11
Remove row 1 and column 1:
| 7 0 2 |
| 1 −1 1 |
| 0 2 1 |
= 7(−1 − 2) + 2(2)
= −21 + 4 = −17
Step 2: Calculation of M13
Remove row 1 and column 3:
| 4 7 2 |
| 1 1 1 |
| 2 0 1 |
= 4(1) − 7(−1) + 2(−2)
= 4 + 7 − 4 = 7
Step 3: Final Calculation
= 1(−17) + (−1)(7)
= −17 − 7 = −30
Correct Answer
Option (B) −30
Conclusion
The value of the determinant of the given 4×4 matrix is −30.
Using the correct expansion method and carefully calculating minors ensures accurate results.
