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

$A^5\left(A{B}^2\right)=A^{5}BA.$

$\Rightarrow B^2=A^{5}BA$

$\Rightarrow B^4=\left(A^{5}BA\right)\left(A^{5}BA\right)=A^5{B}^{2}A=A^5\left(A^{5}BA\right)A$

$\Rightarrow B^4=A^{4}B{A}^2$

$\Rightarrow B^8=\left(A^{4}B{A}^2\right)\left(A^{4}B{A}^2\right)=A^4{B}^2{A}^2=A^4\left(A^{5}BA\right)A^2$

$\Rightarrow B^8=A^{3}B{A}^3$

$\Rightarrow B^{16}=\left(A^{3}B{A}^3\right)\left(A^{3}B{A}^3\right)=A^3{B}^2{A}^3$  = $A^3\left(A^{5}BA\right)A^3=A^{2}B{A}^4$

$A^{32}=\left(A^{2}B{A}^4\right)\left(A^{2}B{A}^4\right)=A^2{B}^2{A}^4=A^2\left(A^{5}BA\right)A^4=AB{A}^5$

$A^{64}=\left(AB{A}^5\right)\left(AB{A}^5\right)=A{B}^2{A}^5$  $=A\left(A^{5}BA\right)A^5=B\Rightarrow A^{63}=I$