[Math] The minimal polynomial divides the characteristic polynomial

Question

How do I show that the minimal polynomial divides the characteristic polynomial? I believe I need to use the Cayley-Hamilton theorem which I understand to be

The characteristic polynomial of a linear operator annihilates it

Answer 1

The Matrix $A$ annihilates the minimal and characteristic polynomial $\mu_A$ and $\chi_A$. We write the Euclidean division of $\chi_A$ by $\mu_A$:

$$\chi_A(x)=\mu_A(x)Q(x)+R(x)$$ with $\deg(R)<\deg(\mu_A)$ so if $R\ne0$ we find a polynomial with degree less than the degree of $\mu_A$ such that $A$ annihilates it and this contradicts the definition of $\mu_A$.