Question

The complex number
$Z=\left|\begin{array}{ccc}2& 3+i& -3\\ 3-i& 0& -1+i\\ -3& -1-i& 4\end{array}\right|$
is equal to

• A. $3-4i$
• B. $5+4i$
• C. $-5i$
• D. $-23$

Answer 1

The correct option is D $-23$

We have
$\overline{z}=\left|\begin{array}{ccc}\overline{2}& \overline{3+i}& -\overline{3}\\ \overline{3-i}& \overline{0}& -\overline{1+i}\\ -\overline{3}& -\overline{1+i}& \overline{4}\end{array}\right|=\left|\begin{array}{ccc}2& 3-i& -3\\ 3+i& 0& -1-i\\ -3& -1+i& 4\end{array}\right|=z$
$=\left|\begin{array}{ccc}2& 3+i& -3\\ 3-i& 0& -1+i\\ -3& -1-i& 4\end{array}\right|=z$ [reflection property
$\therefore z$ is purely real