Question

Let $S=\left\{\begin{array}{c}n\in N|{\left(\begin{array}{cc}0& i\\ 1& 0\end{array}\right)}^n\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\forall a,b,c,d\in R\end{array}\right\}$, where $i=\sqrt{-1}$. Then the number of $2-$digit numbers in the set $S$ is

Answer 1

Let $B=\left(\begin{array}{cc}0& i\\ 1& 0\end{array}\right)$
$\Rightarrow B^2=\left(\begin{array}{cc}i& o\\ 0& i\end{array}\right)$
$\Rightarrow B^4=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$
$\Rightarrow B^8=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
hence $n$ must be a multiple of $8.$
So $n=16,24,32,\dots ,96$
No. of values of $n=11.$