A and B are two singular matrices so that ${\mathrm{A}}^6=\mathrm{I}\text{ and }{\mathrm{AB}}^2=\mathrm{BA}(\mathrm{B}\ne \mathrm{I})$. A value of K so that ${\mathrm{B}}^{\mathrm{K}}=\mathrm{I}$ is

$\begin{array}{l}{\mathrm{A}}^5\left({\mathrm{AB}}^2\right)={\mathrm{A}}^5\mathrm{BA}\\ \Rightarrow {\mathrm{B}}^2={\mathrm{A}}^5\mathrm{BA}\\ \Rightarrow {\mathrm{B}}^4=\left({\mathrm{A}}^5\mathrm{BA}\right)\left({\mathrm{A}}^5\mathrm{BA}\right)={\mathrm{A}}^5{\mathrm{B}}^2\mathrm{A}={\mathrm{A}}^5\left({\mathrm{A}}^5\mathrm{BA}\right)\mathrm{A}\\ \Rightarrow {\mathrm{B}}^4={\mathrm{A}}^4{\mathrm{BA}}^2\\ \Rightarrow {\mathrm{B}}^8=\left({\mathrm{A}}^4{\mathrm{BA}}^2\right)\left({\mathrm{A}}^4{\mathrm{BA}}^2\right)={\mathrm{A}}^4{\mathrm{B}}^2{\mathrm{A}}^2={\mathrm{A}}^4\left({\mathrm{A}}^5\mathrm{BA}\right){\mathrm{A}}^2\\ \Rightarrow {\mathrm{B}}^8={\mathrm{A}}^3{\mathrm{BA}}^3\end{array}$
$\begin{array}{l}\Rightarrow {\mathrm{B}}^{16}=\left({\mathrm{A}}^3{\mathrm{BA}}^3\right)\left({\mathrm{A}}^3{\mathrm{BA}}^3\right)={\mathrm{A}}^3{\mathrm{B}}^2{\mathrm{A}}^3={\mathrm{A}}^3\left({\mathrm{A}}^5\mathrm{BA}\right){\mathrm{A}}^3={\mathrm{A}}^2{\mathrm{BA}}^4\\ {\mathrm{A}}^{32}=\left({\mathrm{A}}^2{\mathrm{BA}}^4\right)\left({\mathrm{A}}^2{\mathrm{BA}}^4\right)={\mathrm{A}}^2{\mathrm{B}}^2{\mathrm{A}}^4={\mathrm{A}}^2\left({\mathrm{A}}^5\mathrm{BA}\right){\mathrm{A}}^4={\mathrm{ABA}}^5\\ {\mathrm{A}}^{64}=\left({\mathrm{ABA}}^5\right)\left({\mathrm{ABA}}^5\right)={\mathrm{AB}}^2{\mathrm{A}}^5=\mathrm{A}\left({\mathrm{A}}^5\mathrm{BA}\right){\mathrm{A}}^5=\mathrm{B}\Rightarrow {\mathrm{A}}^{63}=\mathrm{I}\end{array}$