TITLE: Proving or disproving matrix $A+B$ is invertible
QUESTION [8 upvotes]: Given $A, B \in M_n (\Bbb F)$, where $A$ is $k$-nilpotent and $B$ is invertible, is $A+B$ also invertible?


I was having trouble on how to prove this, and then I thought maybe this statement is incorrect, but couldn't find a counter example. Perhaps someone can assist?

REPLY [9 votes]: A counterexample for $n=2$ by taking
$$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}$$
and 
$$B=\begin{pmatrix}1&0\\1&1\end{pmatrix}$$