Linear Algebra

### **Detailed Step-by-Step Solution (GATE AE 2019 Q.42 / AE Q.32)** **1. Intuitive Physical Setup & Given Parameters**: - Matrix $M = \begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}$. - First eigenvalue: $\lambda_1 = 1$. **2. Step-by-Step Mathematical Derivation**: 1. **Upper Triangular Matrix Property**: Notice that matrix $M$ is an upper triangular matrix because all entries below the main diagonal are zero ($m_{21} = 0$). For any upper triangular (or lower triangular) matrix, the eigenvalues are simply the entries on the main diagonal: $\lambda_1 = m_{11} = 1, \quad \lambda_2 = m_{22} = 2$ 2. **Trace Verification**: The trace of a square matrix equals the sum of its eigenvalues: $\text{Tr}(M) = 1 + 2 = 3$ $\lambda_1 + \lambda_2 = 1 + \lambda_2 = 3 \implies \lambda_2 = 2$ 3. **Determinant Verification**: $\det(M) = (1)(2) - (3)(0) = 2$ $\lambda_1 \lambda_2 = (1)\lambda_2 = 2 \implies \lambda_2 = 2$ **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Exact Evaluation**: **2.0** (Acceptable Range: [2, 2]). **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **2.0** - **Exam Rule**: Eigenvalues of any triangular matrix are its diagonal elements.