Consider this matrix:

$Z=\begin{bmatrix}23.9 & -7 & -17 \\\\ -7 & 23.9 & -17 \\\\ -17 & -17 & 33.9 \end{bmatrix}$

Its inverse is entrywise negative (you can check...) and quite small in absolute value.

Now, the eigenvalues of $Z$ are $-0.1,30.9,50.9$ and if I take the matrix $\widetilde{Z}=Z+(0.1+\epsilon)I$ its inverse flips over into being entrywise positive and very large.

Now, I understand that the poor conditioning of $\widetilde{Z}$ is responsible for the large entries in absolute magnitude - but why the bizarre behaviour of the signs of the entries of $\widetilde{Z}$?

Can you give a conceptual explanation? Is there a reference that targets this very specific issue?