Positivity of Linear equation

Question

When is $A^{-1} b$ positive, given b is non-negative (By non-negativity of a vector I mean that all of its entries are non-negative)? Here $A$ is an $n\times n$ matrix and $b$ is a $n$ dimensional vector. Also we can assume that $A$ is invertible and has all non-negative entries. So what properties of the matrix $A$ forces $A^{-1} b$ to be positive?