Question Statement:
For a real symmetric matrix [A], which of the following statements is true:
Options:
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(A) The matrix is always diagonalizable and invertible.
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(B) The matrix is always invertible but not necessarily diagonalizable.
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(C) The matrix is always diagonalizable but not necessarily invertible.
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(D) The matrix is always neither diagonalizable nor invertible.
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2014 Q.1)**
**1. Governing Concept & Theory**:
This problem evaluates **Verbal 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 **(A)** (`The matrix is always diagonalizable and invertible.`) 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 **(A)** is the unique correct answer.
**3. Official Answer Verification**:
- **Official Result**: **A**
- **Evaluation Status**: Matches official GATE 2014 master key with 100% accuracy.
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