TITLE: Can we deduce the characteristic polynomial for this matrix?
QUESTION [20 upvotes]: Given a square $n \times n$ matrix $A$ that satisfies $$\sum\limits_{k=0}^n a_k A^k = 0$$ for some coefficients $a_0, a_1, \dots, a_n,$ can we deduce that its characteristic polynomial is $\sum\limits_{k=0}^n a_k x^k$?

REPLY [5 votes]: The zero matrix satisfies every homogeneous polynomial, but clearly not every homogeneous polynomial is the characteristic polynomial of the zero matrix.