The value of the determinant \(\begin{vmatrix}1 & 1 & 1\\ 2 & 3 & 5\\ 4 & 9 & 25\end{vmatrix}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 6
The first row is all 1s, so subtracting the first column from the others creates two zeros. Apply \(C_2 \to C_2 - C_1\) and \(C_3 \to C_3 - C_1\); these column operations do not change the value:
\( \displaystyle \Delta = \begin{vmatrix}1 & 0 & 0\\ 2 & 1 & 3\\ 4 & 5 & 21\end{vmatrix} \)
Expanding along the first row:
\( \displaystyle \Delta = 1 \times \begin{vmatrix}1 & 3\\ 5 & 21\end{vmatrix} \)
\( = 21 - 15 \)
\( = 6 \)
Check by direct expansion: \(1(75 - 45) - 1(50 - 20) + 1(18 - 12) = 30 - 30 + 6 = 6\).