The rank after adding a rank-one matrix to a full rank matrix Suppose $A$ and $B$ are all $N$ by $N$ matrices. 
$$rank(A) = N, rank(B) = 1$$
What's the lower bound of:  $$rank(A+B)?$$ 
My guess in this specific case is $$rank(A+B) \geq N-1,$$ but I don't know if it's true, how to prove it, and under what condition we have $rank(A+B) = N-1$.
Can anyone help me with this? I know there is a nice result of the upper bound, $rank(A+B) \leq rank(A) + rank(B)$, but I didn't find anything about the lower bound online.

Note that $rank(X)+rank(Y)\geq rank(X+Y)$, so let $X=A+B$, $Y=-B$, then $rank(A+B)\geq rank(A)-rank(-B)=rank(A)-rank(B)=N-1$
