Linear Algebra

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2020 Q.33)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Matrix Eigenvalues - For matrix $A$, eigenvalues satisfy the characteristic equation $\det(A - \lambda I) = 0$. **2. Detailed Step-by-Step Calculation**: - For the given matrix $A = \begin{pmatrix} 1 & -2 \\ -1 & 2 \end{pmatrix}$: $\operatorname{Tr}(A) = 1 + 2 = 3$ $\det(A) = 1(2) - (-2)(-1) = 2 - 2 = 0$ - Characteristic Equation: $\lambda^2 - \operatorname{Tr}(A)\lambda + \det(A) = 0 \implies \lambda^2 - 3\lambda = 0 \implies \lambda(\lambda - 3) = 0$ - Eigenvalues: $\lambda_1 = 0, \quad \lambda_2 = 3$ - The largest eigenvalue is 3. **3. Final Answer**: - **Calculated Value**: **3** - **Official GATE Accepted Range**: **3 to 3**