Bounding the absolute sum of entries of the inverse of a  0-1 matrix I have a non-singular square 0-1 matrix and I want to bound the sum of absolute values of its inverse as a function of n (or the vector 1-norm).
Asymptotic results are also useful.
Does anyone know any result that can help me?
Thank you,
ifog
 A: Summary The answer below mentions a conjectured lower bound on the Frobenius norm of the inverse of a (0-1)-matrix. I have removed the now irrelevant simple observations that were based on matrices with real entries in $[0,1]$. Exponential upper bounds are discussed in Noam's and David's answers. The average case is described by Terry. 

As far as I know, proving the following lower bound
$$\|A^{-1}\|_F \ge \frac{2n}{n+1},$$
is still an open problem. Further, the conjecture states that this lower bound is achieved iff and only if $A$ is an S-matrix (which is a (0-1)-matrix). See Problem 7 in this handout for more details.
A: An observation: As long as we stick to upper triangular matrices, as in Noam's answer, we can't get growth faster than $2^n$. More precisely, let $a_{ij}$ be an upper triangular $01$ matrix with $1$'s on the diagonal and let $b_{ij}$ be the inverse matrix. Then I claim that $|b_{i(i+k)}| \leq 2^{k-1}$ for all $k>0$.
Proof: Induction on $k$. The case $k=1$ is easy because $b_{i(i+1)} = - a_{i(i+1)}$. In general,
$$\sum_{r=0}^k b_{i(i+r)} a_{(i+r)(i+k)} =0$$
so
$$|b_{i(i+k)}| = \left| \sum_{r=0}^{k-1} b_{i(i+r)} a_{(i+r)(i+k)} \right| \leq  \sum_{r=0}^{k-1} |b_{i(i+r)}|.$$
By induction, the last is bounded by $1+1+2+4+\cdots+2^{k-2} = 2^{k-1}$, and we are done.
A: If one is interested in the typical answer (when the matrix is a random 0-1 matrix) rather than the worst-case answer, then the inverse behaves a lot better than exponential.  Indeed, in view of the results of Rudelson and Vershynin, it is likely that the j^th smallest singular value of the matrix has typical size $j/\sqrt{n}$.  (Technically, the Rudelson-Vershynin result doesn't directly apply because the matrix is not normalised to have mean zero, but it is likely that the conclusions of that paper also apply to the off-centered case, after removing the exceptional outlier singular value of size about n/2.)  Since the Frobenius norm of the inverse is the sum of negative second powers of the singular values, this Frobenius norm should then be about $O(n^{1/2})$, which implies by Cauchy-Schwarz that the $\ell^1$ norm of the inverse should be about $O(n^{3/2})$ typically.   (Roughly speaking, this suggests that individual entries have size $O(n^{-1/2})$, a finding which is consistent with Cramer's rule and the limiting law for the determinant of a random 0-1 matrix (which has value about $\sqrt{(n-1)!}$ on the average, see e.g. this paper of myself and Van Vu). 
A: An answer is given in:
N. Alon and V. H. Vu, Anti-Hadamard matrices, coin weighing,
threshold gates and indecomposable hypergraphs, J. Combinatorial
Theory, Ser. A 79 (1997), 133-160.
Indeed C above is essentially sqrt n
