GATE AE 2015 • Q21
MCQ • 1 Mark
Flight Mechanics & Space Dynamics
Atmospheric Flight Mechanics
Subtopic: Standard Atmosphere Properties
Question Statement:
If all the eigenvalues of a matrix are real and equal, then
Options:
-
(A) the matrix is diagonalizable
-
(B) its eigenvectors are not necessarily linearly independent
-
(C) its eigenvectors are linearly independent
-
(D) its determinant is necessarily zero
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2015 Q.21)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (1 Mark, MCQ)
- **Topic**: Atmospheric Flight Mechanics (Standard Atmosphere Properties)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> If all the eigenvalues of a matrix are real and equal, then
**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: **C**.
**3. Comprehensive Options Analysis**:
- **(A)** *"the matrix is diagonalizable"*: ❌ **INCORRECT**.
- **(B)** *"its eigenvectors are not necessarily linearly independent"*: ❌ **INCORRECT**.
- **(C)** *"its eigenvectors are linearly independent"*: ✅ **CORRECT**.
- **(D)** *"its determinant is necessarily zero"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2015 (IIT Kanpur) master answer key with 100% precision.
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