Cayley Hamilton theorem and divisibility in the reverse direction

Question

Let $A$ be an $n \times n$ matrix with entries in some field $K$. Call $\chi_A$ its characteristic polynomial and $\mu_A$ its minimal polynomial. Cayley Hamilton theorem states that $\mu_A$ divides $\chi_A$. A less known theorem states that $\chi_A$ divides $\mu_A^n$. Does this theorem has a name or/and an identified author?