$AB − BA = A$, $A$ is a nonzero matrix then $B$ is not nilpotent. Let $A,B$ $n\times n$ matrices such that $$AB-BA=A$$ and $A$ is a nonzero matrix. Prove that $B$ is not nilpotent.
I know why $A$ is nilpotent, but how can I prove $B$ is not nilpotent?

If $B$ is nilpotent then there is $k$ such that $AB^k=0$. 
So $-BAB^{k-1}=AB^k-BAB^{k-1}=(AB-BA)B^{k-1}=AB^{k-1}$. 
Hence, $(Id+B)(AB^{k-1})=0$. Since B is nilpotent then $Id+B$ is invertible. Thus, $AB^{k-1}=0$.
We can repeat the argument to prove that $AB^{k-2}=\ldots=AB^1=AB^0=A=0$. Absurd! 
