Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.13)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to find the eigenvalues of the $2 \times 2$ upper triangular matrix $A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$.
**2. Step-by-Step Mathematical Derivation**:
1. **Characteristic Equation Formulation**:
The eigenvalues $\lambda$ satisfy $\det(A - \lambda I) = 0$:
\det \begin{bmatrix} 1 - \lambda & 2 \\ 0 & 3 - \lambda \end{bmatrix} = 0
2. **Expansion of Determinant**:
(1 - \lambda)(3 - \lambda) - (2)(0) = 0
(1 - \lambda)(3 - \lambda) = 0
3. **Roots / Eigenvalues**:
\lambda_1 = 1, \quad \lambda_2 = 3
4. **General Triangular Matrix Theorem**:
For any upper triangular, lower triangular, or diagonal matrix, the characteristic polynomial factors directly into the diagonal entries: $P(\lambda) = \prod_{i=1}^n (a_{ii} - \lambda) = 0$. Hence the eigenvalues are simply the diagonal entries.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 0, 2**: ❌ **INCORRECT**. Confuses the off-diagonal entries with eigenvalues.
- **(B) 2, 3**: ❌ **INCORRECT**. Takes row elements instead of diagonal.
- **(C) 1, 3**: ✅ **CORRECT**. The eigenvalues of an upper triangular matrix are identically its principal diagonal entries $\{1, 3\}$.
- **(D) 1, 2**: ❌ **INCORRECT**. First row entries.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: For any triangular or diagonal matrix of any dimension, you can read the eigenvalues directly from the main diagonal in less than 2 seconds.