Let $F$ be a field and $K/F$ be a finite extension.
For any $x \in K$, there is a minimal polynomial for $x$.
On the other hand, the multiplication by $x$ induces a $F$-linear map $K \to K$. This map has matrix representation, say $A$.
I want to show that $x$ is in fact root of a characteristic polynomial of $A$.
Is this true? And if so, how can I approach this?