If all entries of $A^{-1}$ are positive numbers, then $A$ has the property you desire.
(Edit: This condition is both sufficient and necessary. If one entry $A^{-1}_{ij}$ is negative, then the choice $b=e_j$ will make the $i$-th component of $x=A^{-1}b$ equal to $A^{-1}_{ij}$, but this is negative.)
The following argument shows that your particular example matrix does have this property: your matrix $A$ has the form
$A = 1 - R F$
where $R=\mathrm{diag}(\rho_1,\rho_2,\dots,\rho_n)$ is the diagonal matrix of reflectivities and $F$ the matrix of form factors. The physical interpretation of $R$ and $F$ makes sure that the norm of $RF$ will be smaller than 1, so that $A^{-1}$ can be expressed as a geometric series
$A^{-1} = (1- RF)^{-1} = 1 + RF + (RF)^2 + (RF)^3 + \dots$
Now, since all entries of $RF$ are positive numbers, this formula implies that the entries of $A^{-1}$ are positive numbers as well.