The number of different non-singular matrices of the type $A=\left[\begin{array}{ccc}1& a& c\\ 1& 1& b\\ 0& -w& w\end{array}\right]$  where $w=e^{i\theta }$  and $a,b,c\in \left\{z:z^4-1=0\right\}$  are

$\left|A\right|=w\left(1+b-a-c\right)$

For non singular matrices $\left|A\right|\ne 0$

Total number of different matrices of type $A=4^3=64$

No. of singular matrices : $\left|A\right|=0\Rightarrow 1+b=a+c$

There are only 9 possible cases.

So only 9 matrices are singular $\Rightarrow$  No of non–singular matrices = 64 – 9 = 55