Linear Algebra

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2021 Q.54)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Matrix Determinants - Evaluation of determinant of a $4 \times 4$ patterned cyclic symmetric matrix: $\det(A) = \begin{vmatrix} 1 & 1 & 1 & 1 \\ -1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 \end{vmatrix}$ **2. Detailed Step-by-Step Calculation**: - **Row Operations**: - $R_2 \leftarrow R_2 + R_1$: Row 2 becomes $\begin{pmatrix} 0 & 2 & 2 & 2 \end{pmatrix}$ - $R_3 \leftarrow R_3 - R_1$: Row 3 becomes $\begin{pmatrix} 0 & -2 & 0 & 0 \end{pmatrix}$ - $R_4 \leftarrow R_4 - R_1$: Row 4 becomes $\begin{pmatrix} 0 & 0 & -2 & 0 \end{pmatrix}$ - **Expanding along the first column**: $\det(A) = 1 \cdot \begin{vmatrix} 2 & 2 & 2 \\ -2 & 0 & 0 \\ 0 & -2 & 0 \end{vmatrix}$ - **Expanding along the third column of the $3 \times 3$ matrix**: $\det = 2 \cdot \begin{vmatrix} -2 & 0 \\ 0 & -2 \end{vmatrix} = 2 \cdot [(-2)(-2) - 0] = 2 \cdot 4 = 8$ **3. Final Answer**: - **Calculated Value**: **8** - **Official GATE Accepted Range**: **8 to 8**