Let $A\in M_n(C)$. If $A$ is not a scalar matrix, then prove that there exists an invertible matrix $S$ such that $S^{-1}AS=B$ and $B_{11}=0$

Question

Let $A\in M_n(C)$. If $A$ is not a scalar matrix, then prove that there exists an invertible matrix $S$ such that $S^{-1}AS=B$ and $B_{11}=0$

I have no idea on how to approach this problem. Please help.

Answer 1

Hint 1: You can assume without loss of generality that $A$ is diagonal/Jordan form (why?).

Hint 2: If $A=\begin{bmatrix} a & 0 \\ 0&b \end{bmatrix}$ with $a \neq b$ can you find such an $S$?

This should tell you exactly what to do when $A$ is diagonal or a Jordan form, with not all eigenvalues equal.