Eigenvalues of matrix $A$ $n \times n$, knowing that $A^2+A-6I_n=O_n$ How can I find ALL eigenvalues of matrix $A$, $n \times n$, $n \ge 3$, knowing that 
$$A^2+A-6I_n=O_n.$$
I applied the Cayley-Hamilton theorem, and obviously two of the eigenvalues are $-3$ and $2$.
However, how do I find the rest of them?

I will use a small result related to minimal polynomial. 
Result: The characteristic polynomial and the minimal polynomial have the same roots, possibly with different multiplicities. 
Now, $x^2+x-6=(x+3)(x-2)$ annihilates $A$. So, the minimal polynomial of $A$ divides $(x+3)(x-2)$. So, we have three options for the minimal polynomial : $(x+3)(x-2), (x+3)$ and $(x-2)$. In all the cases we get $-3$ and $2$ are the only possible eigenvalues.
