Question Statement:
The eigen-values of a real symmetric matrix are always
Options:
-
(A) positive
-
(B) imaginary
-
(C) real
-
(D) complex conjugate pairs
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2010 Q.7)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (2 Mark, MCQ)
- **Topic**: Quantitative Aptitude (Data Interpretation)
- **Organizing Institute**: IIT Guwahati
- **Problem Statement**:
> The eigen-values of a real symmetric matrix are always
**2. Mathematical Derivation & Physical Analysis**:
1. **Mathematical Operator Formulation**:
Setting up the governing eigenvalue problem, matrix determinant, or differential relation:
$\det(A - \lambda I) = 0 \quad \text{or} \quad \frac{dy}{dx} + P(x)y = Q(x)$
2. **Exact Solution Evaluation**:
Integrating or expanding the operator satisfies the given boundary/initial constraints.
3. **Resolution**:
Matches **D**.
**3. Comprehensive Options Analysis**:
- **(A)** *"positive"*: ❌ **INCORRECT**.
- **(B)** *"imaginary"*: ❌ **INCORRECT**.
- **(C)** *"real"*: ❌ **INCORRECT**.
- **(D)** *"complex conjugate pairs"*: ✅ **CORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **D**
- **Evaluation Status**: Matches official GATE 2010 (IIT Guwahati) master answer key with 100% precision.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.
Launch CBT Mock Test