Minimal polynomial Let $V$ be the vector space of square matrices of order $n$ over the field $F$. Let $A$ be  a fixed square matrix of $n$ and let $T$ be a linear operator on $V$ such that $T(B) = AB$. Show that the minimal polynomial for $T$ is the minimal polynomial for $A$.
Thank you for your time.

You need to show that every polynomial that kills $A$ kills $T$.  But then to show minimality, you need to show that every polynomial that fails to kill $A$ fails to kill $T$.
$$f(A)=0$$
$$f(T)B = \left(\sum_{k=0}^n c_k T^k \right)B = \sum_{k=0}^n c_k (T^k B).\tag1$$
$$
T^k(B) = T^{k-1}(T(B)) = T^{k-1}(AB) = T^{k-2} (A(AB)) = T^{k-2} (AA(B)) = \cdots.
$$
In other words, show by induction on $k$ that $T^k (B) = A^k B$ and then apply $(1)$.
That should suggest how to do the other part.
