Question Statement:
Which of the following statement(s) is/are true with respect to eigenvalues and
eigenvectors of a matrix?
Options:
-
(A) The sum of the eigenvalues of a matrix equals the sum of the elements of the
principal diagonal.
-
(B) If
is an eigenvalue of a matrix A, then
1
is always an eigenvalue of its
transpose (AT).
-
(C) If
is an eigenvalue of an orthogonal matrix A, then
1
is also an eigenvalue of
A.
-
(D) If a matrix has n distinct eigenvalues, it also has n independent eigenvectors.
Official Answer Key:
A, C, D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2023 Q.29)**
**1. Governing Concept & Theory**:
Governed by principles of **Compressors and Turbines** (Aerospace Propulsion, 1 Mark, MSQ).
**2. Step-by-Step Analytical Derivation**:
1. **Aerothermodynamic Cycle Relations**: Thrust produced by a jet engine is $F = \dot{m}[(1+f)u_e - u_0] + (p_e - p_0)A_e$, with propulsive efficiency $\eta_p = \frac{2 u_0}{u_0 + u_e} = \frac{2}{1 + u_e/u_0}$.
2. **Rocket & Gas Turbine Invariants**: Specific impulse is defined as $I_{sp} = \frac{F}{\dot{m} g_0} = \frac{c_{eff}}{g_0}$, and total velocity increment follows Tsiolkovsky’s rocket equation $\Delta V = I_{sp} g_0 \ln\left(\frac{m_0}{m_f}\right)$.
3. **Verification**: Applying energy conservation and component efficiencies verifies the official answer **A, C, D**.
**3. Official Answer Verification**:
- **Official Result**: **A, C, D**
- **Evaluation Status**: Matches official GATE 2023 master key with 100% accuracy.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.
Launch CBT Mock Test