My problem is this
Let $A$ be an $n\times n$ matrix over $\mathbb{F}$. Prove there are at most $n$ distinct scalars $c\in\mathbb{F}$ such that $\det{(cI-A)}=0.$
I know that the determinant is an $n$-linear function and I was thinking the argument had something to do with that. Then I began to thing that because it's $n$-linear I can facter out a particular constant from each row, but I'm not sure this argument holds any water so to speak. What should be my starting point? Should I assume there are more than $n$ distinct scalars?