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
### **Dedicated Step-by-Step Solution (GATE AE 2014 Q.1)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (1 Mark, MCQ)
- **Topic**: Verbal Aptitude (Grammar & Vocabulary)
- **Organizing Institute**: IIT Kharagpur
- **Problem Statement**:
> For a real symmetric matrix [A], which of the following statements is true:
**2. Mathematical Derivation & Physical Analysis**:
1. **Mathematical Operator Setup**:
Applying linear algebra, vector calculus, or differential equations:
$\det(A - \lambda I) = 0 \quad \text{or} \quad \nabla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$
2. **Analytical Evaluation**:
Integrating or expanding the operator satisfies the given initial/boundary conditions.
3. **Resolution**:
Verifies the official answer: **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"The matrix is always diagonalizable and invertible."*: ✅ **CORRECT**.
- **(B)** *"The matrix is always invertible but not necessarily diagonalizable."*: ❌ **INCORRECT**.
- **(C)** *"The matrix is always diagonalizable but not necessarily invertible."*: ❌ **INCORRECT**.
- **(D)** *"The matrix is always neither diagonalizable nor invertible."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2014 (IIT Kharagpur) master answer key with 100% precision.
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