GATE BT 2022 • Q12
MCQ • 1 Mark
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Fundamentals of Linear Algebra
Question Statement:
If the eigenvalues of a 2×2 matrix P are 4 and 2, then the eigenvalues of the
matrix
1
P are
Options:
-
(A) 0, 0
-
(B) Q.0625, 0.25
-
(C) Q.25, 0.5
-
(D) 2, 4
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2022 Q.12)**
**1. Governing Concept**:
- **Topic**: Linear Algebra & Matrix Properties (Eigenvalues)
- Fundamental spectral theorem properties for any square matrix $P$:
1. The **trace** of matrix $P$ equals the sum of its eigenvalues:
$\text{Tr}(P) = \lambda_1 + \lambda_2 = 4 + 2 = 6$
2. The **determinant** of matrix $P$ equals the product of its eigenvalues:
$\det(P) = \lambda_1 \cdot \lambda_2 = 4 \times 2 = 8$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Incorrect trace/determinant combination.
- **Option (B) is INCORRECT**: Does not satisfy eigenvalue products.
- **Option (C) is CORRECT**: Matches $\text{Tr}(P) = 6$ and $\det(P) = 8$.
- **Option (D) is INCORRECT**: Incompatible determinant.
**3. Final Answer**:
- **Correct Choice**: **C**
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