Let $J_k$ be a $k \times k$ all ones matrix and $B$ **any** $k \times k$ binary matrix - that is $B$ only has entries from $\{0,1\}$.

I would like to show that the matrix $$X_B = (J_k -I) - B (J_k - I)^{-1} B^T\,,$$ is not positive-definite. In other words, I'd like to show that

At least one eigenvalue of $X_B$ is non-positive.

I can show that for certain specific matrices $B$ but don't see how to prove the more general statement. Does anybody know why this property seem to hold in general?