Suppose $A$ and $B$ are two non-singular matrices such that $A \neq I$, $B^4 = I$, and $BA^3 = AB$, then what is the least value of $k$ for which $A^k = I$?
\begin{align}k&=80\end{align}
My attempt:
By taking inverse in given relation,$A^3 = B^{-1}AB$
Since $B^4=I$, $B^{-1}=B^3$. So, $A^3=B^3AB$. How to proceed further?
Starting from what you got: $$ A^3 = B^{-1} A B $$ Raise to the 3rd power: $$ A^9 = B^{-1} A^3 B = B^{-2} A B^2 $$ Doing it two more times: $$ A^{81} = B^{-4} A B^4 = A $$ Cancelling $A$: $$ A^{80} = I $$ I skipped some of the details but I'm sure you can fill the blanks.