Can we deduce the characteristic polynomial for this matrix? 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 $\sum\limits_{k=0}^n a_k x^k$? 

Take matrix
$$
A = 
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 2 \\
\end{array}
\right)
$$
Then
$$  A^4 - 11 A^3 + 41 A^2 - 61 A + 30 I = 0$$
However, the characteristic polynomial is
$$  x^4 - 6x^3 + 13x^2 - 12x + 4 $$
and the minimal polynomial is
$$ x^2 - 3 x + 2 $$
