Question

Let $A$ and $B$ be two non-singular matrices such that $A\ne I,B^3=I$ and $AB=B{A}^2$, find the least value of $k$ such that $A^k=I$.

Answer 1

$B^3=I,$ $AB=B{A}^2$

$\Rightarrow B=A^{-1}B{A}^2$

$B^2=A^{-1}B{A}^2{A}^{-1}B{A}^2$

$=A^{-1}BAB{A}^2$

$=A^{-1}BB{A}^2{A}^2$

$=A^{-1}{B}^2{A}^4$

$I=B^3=A^{-1}{B}^2{A}^4{A}^{-1}B{A}^2$

$=A^{-1}{B}^2{A}^{3}B{A}^2$

$=A^{-1}{B}^2{A}^{2}B{A}^2{A}^2$

$=A^{-1}{B}^{2}AAB{A}^4$

$=A^{-1}{B}^{2}AB{A}^2{A}^4$

$=A^{-1}{B}^{2}B{A}^2{A}^6$

$=A^{-1}{B}^3{A}^8=A^7$ ........ $[\because B^3=I]$

$\therefore k=7$