$AB − BA = A$, $A$ 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?

Here's one way to see it: define the linear transformation
$$
\Phi_B(X) = XB - BX
$$
clearly, $A$ is an eigenvector of this transformation associated with $\lambda = 1$.  Now, the eigenvalues of $\Phi_B$ are necessarily of the form $\lambda - \mu$ where $\lambda,\mu$ are eigenvalues of $B$ (can be seen via vectorization).  
Conclude that since $\Phi_B$ has a non-zero eigenvalue, $B$ has a non-zero eigenvalue, which means it is not nilpotent.
