matrices - How can I characterize the type of solution vector that comes out of a matrix?

If and only if all the entries of $A^{-1}$ are non-negative.

Proof: If $(A^{-1})_{ij}$ is negative, and $b$ is $1$ in the $j$-th coordinate and very small in every other, then $A^{-1} b$ is negative in the $i$-th component.

On the other hand, if every entry of $A^{-1}$ is non-negative, then clearly $b$ positive implies $A^{-1} b$ positive.