The value of […]

The correct answer is Option (3) → 0

To evaluate the determinant of the matrix:

$ \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \end{vmatrix} $

Step 1: Expand along the first row

$ D = 265 \cdot \begin{vmatrix}225 & 198 \\ 198 & 181\end{vmatrix} - 240 \cdot \begin{vmatrix}240 & 198 \\ 219 & 181\end{vmatrix} + 219 \cdot \begin{vmatrix}240 & 225 \\ 219 & 198\end{vmatrix} $

Step 2: Evaluate each 2×2 determinant

$ \begin{vmatrix}225 & 198 \\ 198 & 181\end{vmatrix} = 225 \cdot 181 - 198 \cdot 198 = 40725 - 39204 = 1521 $

$ \begin{vmatrix}240 & 198 \\ 219 & 181\end{vmatrix} = 240 \cdot 181 - 198 \cdot 219 = 43440 - 43362 = 78 $

$ \begin{vmatrix}240 & 225 \\ 219 & 198\end{vmatrix} = 240 \cdot 198 - 225 \cdot 219 = 47520 - 49275 = -1755 $

Step 3: Plug in all values

$ D = 265 \cdot 1521 - 240 \cdot 78 + 219 \cdot (-1755) $

$ D = 403065 - 18720 - 384645 = 0 $