Counterexample involving the minimal polynomial of a linear operator

Question

Let $F$ be a field and let $V = F^{4\times4}$ be the vector space of $4x4$ matrices over $F$. For $A \in F^{4\times4}$, define $T_A : V \rightarrow V$ by $T_A(B) = AB$ for each $B \in V$.

Question: True or false "The minimal polynomial of $T_A$ is never equal to the characteristic polynomial of $T_A$"

I know how to prove that the minimal polynomial of $T_A$ is equal to the minimal polynomial of $A$. I was thinking I could pick a suitable diagonal matrix $A$ to make the question false but after testing a couple of examples I am failing to come up with the correct diagonal matrix.

Answer 1

I know how to prove that the minimal polynomial of $T_A$ is equal to the minimal polynomial of $A$.

Good, then you are almost done. The minimal polynomial of $A$ has degree at most [blank], while the characteristic polynomial of $T_A$ has degree equal to [blank]. You can fill in the blanks by looking at the dimensions of the spaces which $A$ and $T_A$ act on.