Definition: Let $A$ be a ring and $Z=Z(A)$ its center. We say that $t \in A$ is algebraic over $Z$ if there exist $z_0,z_1, \ldots , z_n \in Z$ such that $$z_0+z_1t+ \cdots + z_n t^n = 0 \quad \text{and} \quad z_n \neq 0.$$
Moreover, in this case we say that $t$ is algebraic of degree at most $n$. The degree of algebraicity is the smallest $n$ that satisfies this property; it will be denoted by $\text{deg}(t)$.
Show that $\text{deg}(M_n(\mathbb{K})) = n$, where $\mathbb{K}$ is a field.
Comments: I'm trying to use the Cayley-Hamilton's Theorem.