Question

If $\Delta$ = $\left|\begin{array}{ccc}a& s-i& 7+i\\ s+i& b& 3+i\\ 7-i& 3-i& c\end{array}\right|$ , then $\Delta$ is always

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

Answer 1

The correct option is A real

$\overline{\Delta }=\left|\begin{array}{ccc}a& s+i& 7-i\\ s-i& b& 3-i\\ 7+i& 3+i& c\end{array}\right|$

$\Delta -\overline{\Delta }=\left|\begin{array}{ccc}0& -2i& 2i\\ 2i& 0& 2i\\ -2i& -2i& 0\end{array}\right|=0$

$\Delta -\overline{\Delta }=0$
$\therefore \Delta =\overline{\Delta }$ is real
$\ast$ note : if a complex number is purely real then $\Delta =\overline{\Delta }$.

Hence, $\Delta$ is real.