Linear Algebra

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2025 Q.63)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Matrix Eigenvalues - For any square matrix $A$, the trace of the matrix is equal to the sum of its eigenvalues: $\operatorname{Tr}(A) = \sum_{i=1}^n \lambda_i$ **2. Detailed Step-by-Step Calculation**: - **Given Matrix**: $A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}$ - **Given Eigenvalues**: $\lambda_1 = -2, \quad \lambda_2 = 1, \quad \lambda_3 = 2$ - **Step 1: Compute trace from diagonal elements**: $\operatorname{Tr}(A) = 1 + k + (-1) = k$ - **Step 2: Compute trace from sum of eigenvalues**: $\sum \lambda_i = (-2) + 1 + 2 = 1$ - **Step 3: Equating trace expressions**: $k = 1$ **3. Final Answer**: - **Calculated Value**: **1** - **Official GATE Accepted Range**: **1 to 1**