Question

If $x,y,z$ are complex numbers, and
$\Delta =\left|\begin{array}{ccc}0& -y& -z\\ \bar{y}& 0& -x\\ \bar{z}& \bar{x}& 0\end{array}\right|$ then $\Delta$ is

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

Answer 1

Answer: (B)

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