If a, b, c are complex numbers, then the determinant

$Δ=\begin{vmatrix}0 & -b & -c\\\overline{b} & 0 & -1\\\overline{c} & \overline{a} & 0\end{vmatrix},$ is

purely imaginary

The correct answer is option (2) : purely imaginary

We observe that

$\overline{Δ}=\begin{vmatrix}0 & -\overline{b} & -\overline{c}\\b & 0 & -\overline{a}\\\overline{c} &a & 0\end{vmatrix}$

$⇒\overline{Δ}=-\begin{vmatrix}0 & \overline{b} & \overline{c}\\-b  & 0 & \overline{a}\\-c & -a & 0\end{vmatrix}$ [Taking )-1) common fromeach row]

$⇒\overline{Δ}=-\begin{vmatrix}0 & -b & -c\\\overline{b} & 0 & -a\\\overline{c} & \overline{a} & 0\end{vmatrix}$ [Interchanging rows and columns]

$⇒\overline{Δ}=-Δ⇒Δ+\overline{Δ}=0⇒Δ $ is purely imaginary.