If $\mathrm{A}=\left[\begin{array}{cc}\mathrm{i}& -\mathrm{i}\\ -\mathrm{i}& \mathrm{i}\end{array}\right]\text{ and }\mathrm{B}=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, then A⁸ = k B , then k=

We have,

A = iB

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

or ${\mathrm{A}}^4=(-2\mathrm{B}{)}^2=4{\mathrm{B}}^2=4(2\mathrm{B})=8\mathrm{B}$

or ${\left({\mathrm{A}}^4\right)}^2=(8\mathrm{B}{)}^2$

or ${\mathrm{A}}^8=64{\mathrm{B}}^2=128\mathrm{B}$