Let  $A$ is a four order diagonal matrix whose entries are  real or complex numbers such that $A^4=I_4.$ If trace of $A$ is zero then which is/are  correct? (here  $I_4$ is four order unit matrix)

$$\begin{gathered} A_{4\times 4}=\left(\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& d\end{array}\right)\&\text{\hspace{0.17em}trace\hspace{0.17em}of\hspace{0.17em}}A=0 \\ A^4=\left(\begin{array}{cccc}{a}^4& 0& 0& 0\\ 0& b^4& 0& 0\\ 0& 0& c^4& 0\\ 0& 0& 0& d^4\end{array}\right)=I\&a+b+c+d=0 \\ \therefore a^4=1\&b^4=1\&c^4=1\&d^4=1\&a+b+c+d=0 \\ a,b,c,d\in \{1,-1,i,-i\}a+b+c+d=0 \\ a,b,c,d \left(i\right)Useall1,-i,i,-i\Rightarrow 4!=24 \\ \left(ii\right) \text{\hspace{0.17em}Use\hspace{0.17em}all\hspace{0.17em}}1,-1,1,-1\Rightarrow \frac{4!}{2!2!}=6 \\ \left(iii\right) \text{\hspace{0.17em}Use\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}}i,-i,i,-i\Rightarrow \frac{4!}{2!2!}=6 \\ R.N.W = 36 \end{gathered}$$