Let $A\in\{0,1\}^{n\times n}$ be a 0-1 matrix. It may be considered as a matrix over various fields. Let $p$ be a prime. Assume A is invertible over $\mathbb{F}_p$, the finite field with $p$ elements. I believe that in this case $A$ is also invertible over $\mathbb{Q}$ and $\mathbb{R}$ (by considering the determinant, for instance).
My question is: Is there a way to retrieve the inverse of $A$ over $\mathbb{R}$ (or $\mathbb{Q}$) explicitly from the inverse of $A$ over $\mathbb{F}_p$?
Using $A^{-1} = \frac{1}{\det A} \operatorname{adj}(A)$, it suffices to retrieve $\operatorname{adj}(A)$ over $\mathbb Q$ from $\operatorname{adj}(A)$ over $\mathbb F_p$.
The entries of $\operatorname{adj}(A)$ certainly lie between $-\frac{(n-1)!}{2}$ and $\frac{(n-1)!}{2}$, so just have to pick $p>(n-1)!$ and you have that $\operatorname{adj}(A)$ over $\mathbb F_p$ and $\mathbb Q$ coincide, if you choose the representatives between $-\frac{p}{2}$ and $\frac{p}{2}$.