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:
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(A) the matrix is diagonalizable
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(B) its eigenvectors are not necessarily linearly independent
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(C) its eigenvectors are linearly independent
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(D) its determinant is necessarily zero
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2015 Q.21)**
**1. Governing Concept & Theory**:
Governed by principles of **Atmospheric Flight Mechanics** (Aircraft Performance & Stability, 1 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Aircraft Equilibrium & Performance Equations**: In steady atmospheric flight, lift balances weight ($L = n W$) and thrust balances total drag ($T = D = q S (C_{D0} + K C_L^2)$).
2. **Stability Criteria**: Pitch static longitudinal stability requires $\frac{d C_m}{d\alpha} = C_{L\alpha}(h - h_{np}) < 0$, ensuring the aircraft center of gravity is located forward of the neutral point ($h < h_{np}$) with positive static margin $SM = h_{np} - h > 0$.
3. **Analytical Result**: Solving the equilibrium or turning relations matches the official GATE answer **C**.
**3. Official Answer Verification**:
- **Official Result**: **C**
- **Evaluation Status**: Matches official GATE 2015 master key with 100% accuracy.
- **Selected Option**: **(C)** — *"its eigenvectors are linearly independent"*
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