Rank of a matrix shifted by all one matrix over any field

Question

Let $J$ be all one square matrix. Is it true that over any field, for all square matrices $A$, $$|\operatorname{rank}(J\pm A)-\operatorname{rank}(A)|\leq c$$ for some positive constant $c$?

Answer 1

Hint. Let $A_1, \dots, A_n$ be column vectors, and let $E$ be a vector consisting of all ones. Prove that the rank of the system $(A_1 + E, \dots, A_n + E)$ is at most one more than the rank of the system $(A_1, \dots, A_n)$.

Answer 2

answers is $c = 1.$ here is why: your all one matrix $J = aa^T$ where $a$ is the vector with one for all components is of rank one. now use the fact that $rank(A \pm B) \le rank(A) + rank (B),$ with $B = J.$