Complex Matrix and its Conjugate terms If $z = \left| {\begin{array}{*{20}{c}}
{3 + 2i}&1&i\\
2&{3 - 2i}&{1 + i}\\
{1 - i}&{ - i}&3
\end{array}} \right|\& \left| {z + \overline z } \right| = k\left| z \right|$, find the value of k
My approach is as follow
$ \Rightarrow z =  - \left| {\begin{array}{*{20}{c}}
1&{3 + 2i}&i\\
{3 - 2i}&2&{1 + i}\\
{ - i}&{1 - i}&3
\end{array}} \right|$ where $\left( {{C_1} \leftrightarrow {C_2}} \right)$
$ \Rightarrow \overline z  =  - \left| {\begin{array}{*{20}{c}}
1&{3 - 2i}&{ - i}\\
{3 + 2i}&2&{1 - i}\\
i&{1 + i}&3
\end{array}} \right|$
How do we proceed from here.

Since $z=27$, $\left|z+\overline z\right|=2\times27$, and therefore $k=2$.
