Question

If $\Delta =\left|\begin{array}{ccc}a& 5-i& 7+i\\ 5+i& b& 3+i\\ 7-i& 3-i& c\end{array}\right|$, then $\Delta$ is always

• A. real
• B. imaginary
• C. $0$
• D. None of these

Answer 1

The correct option is A real

$\Delta =\left|\begin{array}{ccc}a& 5-i& 7+i\\ 5+i& b& 3+i\\ 7-i& 3-i& c\end{array}\right|$
$=abc-a(9-i^2)+(i-5)(5c+ic-(7-i\left)\right(3+i\left)\right)+(7+i)\left[\right(5+i\left)\right(3-i)-b(7-i\left)\right]$
$=abc-10a+5ic-c-25c-5ic-(i-5)(21+4i+1)+(7+i)(16-2i)-50b$
$=abc-10a-26c-2i+114+114+2i-50b$
$\Delta =abc-10a-26c+228-50b$
Hence, $\Delta$ is real.