Question

Without expanding the determinant at any stage, $\left|\begin{array}{ccc}-5& 3+5i& \frac{3}{2}-4i\\ 3-5i& 8& 4+5i\\ \frac{3}{2}+4i& 4-5i& 9\end{array}\right|$,
its value is

• A. a purely real
• B. purely complex
• C. complex
• D. zero

Answer 1

The correct option is A a purely real

Let $z=\left|\begin{array}{ccc}-5& 3+5i& \frac{3}{2}-ri\\ 3-5i& 8& 4+5i\\ \frac{3}{2}+4i& 4-5i& 9\end{array}\right|$
then $\overline{z}=\left|\begin{array}{ccc}-5& 3-5i& \frac{3}{2}+4i\\ 3+5i& 8& 4-5i\\ \frac{3}{2}-4i& 4+5i& 9\end{array}\right|$ (i.e. conjugate of z)
$=\left|\begin{array}{ccc}-5& 3+5i& \frac{3}{2}-4i\\ 3-5i& 8& 4+5i\\ \frac{3}{2}+4i& 4-5i& 9\end{array}\right|$
(Interchanging rows into columns)
$=z$
Hence $\overline{z}=z$
Let $z=a+ib$
then $a-ib=a+ib$
$\therefore b=0$
$z=a+0$ $=a\Rightarrow$ purely real.