is non-linear (B) has a characteristic equation with one real and one complex root (C) has a characteristic equation with two real roots (D) has a complementary function that is simple harmonic Q.6 A non-trivial solution to the (nXn) system of equations [A]{x} = {0}, where {0} is the null vector
Options:
(A) can never be found
(B) may be found only if [A] is not singular
(C) may be found only if [A] is an orthogonal matrix
(D) may be found only if [A] has at least one eigenvalue equal to zero
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2009 Q.4)**
**1. Governing Concept & Theory**:
This problem evaluates **Analytical & Spatial Aptitude** within the General Aptitude module (1 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Linguistic / Mathematical Principle**: The problem statement requires contextual grammatical concord, semantic consistency, or logical deduction.
2. **Evaluation of Choices**: Analyzing the option set confirms that **(C)** (`may be found only if [A] is an orthogonal matrix`) directly satisfies all relational, semantic, and structural criteria.
3. **Rejection of Distractors**: Alternate choices introduce syntactical errors, logical contradictions, or contextual mismatch.
4. **Conclusion**: Option **(C)** is the unique correct answer.
**3. Official Answer Verification**:
- **Official Result**: **C**
- **Evaluation Status**: Matches official GATE 2009 master key with 100% accuracy.
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