$A^n$ in terms of change-of-basis S

Question

So I am given two basis, one is the standard basis $E$ and the other is another basis $B$.

I have found the change-of-matrices $S_{E→B}$ and $S_{B→E}$, and I want to compute $A^n$ in terms of $S_{E→B}$ and $S_{B→E}$ for each $n ∈ N$,

I know that $B$ and $A$ are similar matrices, and $A_{[B]}$ = $S_{B→E}A_{[E]}S_{E→B}$

so I think A = $A_{[B]} * B$ = $S_{B→E}A_{[E]}S_{E→B}B$

But I am stucked here , waht should I do (next) to find $A$ in terms of $S_{E→B}$ and $S_{B→E}$, so that I could find $A^n$?

Answer 1

Hint: $$ S_{E\to B} =S \Rightarrow S_{B\to E}=S^{-1} $$ so: $$ A^2=\left(SA_ES^{-1}\right)^2=SA_ES^{-1}SA_ES^{-1}=SA_E^2S^{-1} $$

Now use induction.