If \(f(x) = \begin{vmatrix} 0 & x-1 & x-2 \\ x+1 & 0 & x-3 \\ x+2 & x+3 & 0 \end{vmatrix}\), then the value of \(f(0)\) is equal to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0
Putting x = 0:
\( f(0) = \begin{vmatrix} 0 & -1 & -2 \\ 1 & 0 & -3 \\ 2 & 3 & 0 \end{vmatrix} \)
Expanding along the first row:
\(f(0) = 0\,(0 + 9) - (-1)(0 + 6) + (-2)(3 - 0)\)
\(= 0 + 6 - 6 = 0\)
Shortcut: The matrix of f(0) has zero diagonal and \(a_{ij} = -a_{ji}\), so it is skew-symmetric of order 3. The determinant of a skew-symmetric matrix of odd order is always 0.