If A = $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right] and B=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$ the A⁸ equals

We have, 

$A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]=i \left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]= i B$

$\Rightarrow A^2={\left(iB\right)}^2=i^2{B}^2=-B^2=-\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$

$=\left[\begin{array}{cc}-2& 2\\ 2& -2\end{array}\right]=-2B$

⇒ A⁴ = (–2B)² = 4^B2 = 4(2B) = 8B
⇒ A⁸ = (A⁴)² = (8B)² = 64B² = 128B