Question: Consider a $n \times n$ matrix $A=(a\_{ij})$ with $a\_{12}=1$, $a\_{ij} =0 \ \forall \ (i,j) \neq (1,2)$. Prove that there is no invertible matrix $P$ such that $PAP^{-1}$ is a diagonal matrix. 

Any help?

Answer: 
If there's some invertible $P$ such that
$$PAP^{-1}=D$$ 
is diagonal, then 
$$PA^2P^{-1}=D^2$$ 
$$\Rightarrow P0P^{-1}=D^2$$
$$\Rightarrow D=0$$
So
$$A=0$$
a contradiction.
Observe that $A$ is upper triangular, so that you can read off the eigenvalues of $A$. What are the eigenvalues of $A$, then? What would this fact imply about $PAP^{-1}$ if such a $P$ existed?