If a, b, c are complex numbers, then the determinant $Δ=\begin{vmatrix}0&-b&-c\\\bar b&0&-a\\\bar c&\bar a&0\end{vmatrix}$, is 

purely imaginary

We observe that

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

$⇒\overline{Δ}=-\begin{vmatrix}0&\overline b&\overline c\\-b&0&\overline a\\-c&-a&0\end{vmatrix}$  [Taking (-1) common from each 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.