GATE BT 2026 • Q12
MCQ • 1 Mark
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Fundamentals of Linear Algebra
Question Statement:
Which of the following is one of the eigenvalues for the matrix given below?
[3 4
4 -3]
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.12)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Matrix Eigenvalues
- For matrix $M = \begin{pmatrix} 3 & 4 \\ 4 & 3 \end{pmatrix}$:
- Characteristic Equation:
$\det(M - \lambda I) = 0 \implies (3 - \lambda)^2 - 4^2 = 0$
$(3 - \lambda - 4)(3 - \lambda + 4) = 0 \implies (-\lambda - 1)(7 - \lambda) = 0$
$(\lambda + 1)(\lambda - 7) = 0 \implies \lambda_1 = 7, \quad \lambda_2 = -1$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Not an eigenvalue.
- **Option (B) is INCORRECT**: Not an eigenvalue.
- **Option (C) [7] is CORRECT**: Matches the dominant positive eigenvalue $\lambda = 7$.
- **Option (D) is INCORRECT**: Not an eigenvalue.
**3. Final Answer**:
- **Correct Choice**: **C**
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