Linear algebra, question about polynoms A,B are matrices n*n over a field F.
I am given a polynom f(t) {belongs to F[t]} . How can I show that Af(BA)B= ABf(AB)?
I defined a polynom g(t)= t*f(t). Then I substituted AB instead of t, but I cant find a way to continue.
Thanks!

If $f(t)=t^k$, then
$$Af(BA)B=A\underbrace{BABA\cdots BA}_{k\,\,\text{times}}B=ABf(AB)$$
Now if the claim is true for polynomials $f,g$ then it is also true for $f+g$. The same goes for scalar multiples.
Since every polynomial is a linear combination of monomials, the claim is proven.
