Upper bounds on the approximate inverse of a singular matrix Let $A$ be a singular matrix and $\zeta>0$ such that $A+\zeta I$ is nonsingular. 
Soft question: Is it true that if $\zeta$ is sufficiently small, then $$(A+\zeta I)^{-1}A \approx I$$ (or $A(A+\zeta I)^{-1} \approx I$)? 
More precisely: Let $E:=(A+\zeta I)^{-1}A - I$. If the answer to the above (soft) question is "yes", what are some known bounds on $E$?
Note. Using the condition number $\kappa(A)$, a well-known bound is 
$$
\frac{\Vert(A+B)^{-1} - A^{-1}\Vert}{\Vert A^{-1}\Vert}
\le \kappa(A)\frac{\Vert B\Vert}{\Vert A\Vert },
$$
but this bound requires that both $A$ and $A+B$ are nonsingular. In my question, $A$ is singular while $A+B$ is nonsingular.

Try some trivial test cases, like
$$
A=\pmatrix{1&0\\0&0}\implies A(A+\zeta I)^{-1}=\pmatrix{(1+ζ)^{-1}&0\\0&0}
$$
which is nowhere close to the identity matrix.
