Let A and B are two nonsingular matrices such that ${\mathrm{A}}^6=\mathrm{I} \text{and }{\mathrm{AB}}^2=\mathrm{BA}.$ If the least value of m such that ${\mathrm{B}}^{\mathrm{m}}=\mathrm{I}(\mathrm{B}\ne \mathrm{I})(\mathrm{m}\in \mathrm{N}) is K$ then the value of $K$ 

${\mathrm{AB}}^2=\mathrm{BA}\Rightarrow {\mathrm{B}}^2={\mathrm{A}}^{-1}\mathrm{BA} \text{squaring both the sides}$

${\left({\mathrm{B}}^2\right)}^2=\left({\mathrm{A}}^{-1}\mathrm{BA}\right)\left({\mathrm{A}}^{-1}\mathrm{BA}\right)={\mathrm{A}}^{-1}{\mathrm{B}}^2\mathrm{A}$

$B^4={\mathrm{A}}^{-1}\left({\mathrm{A}}^{-1}\mathrm{BA}\right)\mathrm{A}$

${\mathrm{B}}^4={\left({\mathrm{A}}^{-1}\right)}^2{\mathrm{BA}}^2$

Again squaring

${\mathrm{B}}^8=\left({\left({\mathrm{A}}^{-1}\right)}^2{\mathrm{BA}}^2\right)\left({\left({\mathrm{A}}^{-1}\right)}^2{\mathrm{BA}}^2\right)$

$\Rightarrow {\mathrm{B}}^8={\left({\mathrm{A}}^{-1}\right)}^2{\mathrm{B}}^2{\mathrm{A}}^2={\left({\mathrm{A}}^{-1}\right)}^2\left({\mathrm{A}}^{-1}\mathrm{BA}\right){\mathrm{A}}^2$

$\Rightarrow {\left({\mathrm{A}}^{-1}\right)}^3{\mathrm{BA}}^3$

Similarly

${\mathrm{B}}^{64}={\left({\mathrm{A}}^{-1}\right)}^0{\mathrm{BAA}}^6\Rightarrow {\mathrm{B}}^{63}=1$

$\text{so}m=63=k$