On a question about nilpotent matrices

Question

Let $A,B\in M_n(\mathbb C)$ be arbitrary matrices and $A,B$ isn't nilpotent. Prove that $A+sB$ can not be nilpotent for infinite $s\in \mathbb C$.

I tried to show that: $trace((A+sB)^k)=0$ for every $k\in \mathbb N$, can not happen for infinite $s\in \mathbb C$ but i stick here and now I don't have any idea for this question. can somebody help me please?

Tanks in advance...

Answer 1

Possible hint: a square matrix is nilpotent only if its characteristic polynomial is $\lambda ^n$.

Answer 2

Hint: if $A\in M_n(\mathbb C)$ is nilpotent, then $A^n=0$.