Question

If $z=\left|\begin{array}{ccc}-5& 3+4i& 5-7i\\ 3-4i& 6& 8+7i\\ 5+7i& 8-7i& 9\end{array}\right|$, then $z$

• A. purely real
• B. purely imaginary
• C. $a+ib$, where $a\ne 0,b\ne 0$
• D. $a+ib$, where $b=4$

Answer 1

The correct option is A purely real

$z=\left|\begin{array}{ccc}-5& 3+4i& 5-7i\\ 3-4i& 6& 8+7i\\ 5+7i& 8-7i& 9\end{array}\right|$
Given matrix is a Hermitian matrix.
We know that determinant of Hermitian matrix is a real number.
Hence, option 'A' is correct.