Mathematics - Matrix Question with Solution | TestHub

$A^5\left(A{B}^2\right)=A^{5}BA$. $\Rightarrow B^2=A^{5}BA$ $\Rightarrow B^4=\left(A^{5}BA\right)\left(A^{5}BA\right)=A^5{B}^{2}A=A^5\left(A^{5}BA\right)A$ $\Rightarrow B^4=A^{4}B{A}^2$ $\Rightarrow B^8=\left(A^{4}B{A}^2\right)\left(A^{4}B{A}^2\right)=A^4{B}^2{A}^2=A^4\left(A^{5}BA\right)A^2$ $\Rightarrow B^8=A^{3}B{A}^3$

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