GATE AE 2014 • Q1 MCQ • 1 Mark General Aptitude

General Aptitude

Quantitative Aptitude

Subtopic: Arithmetic & Numerical Estimation

Question Statement:

For a real symmetric matrix [A], which of the following statements is true:

Options:

  • (A) The matrix is always diagonalizable and invertible.
  • (B) The matrix is always invertible but not necessarily diagonalizable.
  • (C) The matrix is always diagonalizable but not necessarily invertible.
  • (D) The matrix is always neither diagonalizable nor invertible.

Official Answer Key:

A

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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