Question

If $x, y, z$ are complex numbers, and
$\Delta = \begin{vmatrix}0&-y&-z\\ \bar{y} &0&-x\\ \bar{z}&\bar{x}&0\end{vmatrix}$ then $ \Delta$ is

• A. purely real
• B. purely imaginary
• C. complex
• D. 0

Answer 1

Answer: (B)

We have
$\bar{\Delta} = \begin{vmatrix}0&- \bar{y}& - \bar{z}\\ y &0& -\bar{x}\\ z&x&0\end{vmatrix} $
[Interchanging rows and columns]
$= \begin{vmatrix}0&y&z\\ -\bar{y}&0&x\\ - \bar{z}& -\bar{x}&0\end{vmatrix} = \left(-1\right)^{3} \begin{vmatrix}0&-y&-z\\ \bar{y}&0&-x\\ \bar{z}&\bar{x}&0\end{vmatrix} = -\Delta$
[Taking -1 common from each row]
$\therefore \bar{\Delta } + \Delta = 0 \Rightarrow 2 Re \left(\Delta\right) = 0 $
$\therefore \Delta$ is purely imaginary.