$A\in \mathbb{R}^{n\times n}$ is an n-by-n matrix. $x=(x_1,x_2,\ldots ,x_n)\in \mathbb{R}^n$ is a vector. $x\geq 0$ means $x_i\geq 0,\forall i$.
Q1: When $A$ satisfies what conditions, $\forall x\in\mathbb{R}^n ,x\geq 0\Rightarrow Ax\geq 0$?
Q2: When $A$ satisfies what conditions, $\forall x\in\mathbb{R}^n ,Ax\geq 0\Rightarrow x\geq 0$?
Q1 is solved by Martin Argerami(The matrix's entries are all positive). But what are the matrices in Q2?
In Q2, the answer is not "entries all positive". The counterexample is $A=\begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}$, $A\begin{pmatrix} -1 \\ 2 \end{pmatrix}\geq 0$.
Let's see an example $A=\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$.$A\begin{pmatrix} x \\ y \end{pmatrix} \geq 0\Rightarrow x\geq 0,y\geq 0$. So $A$ is in answer to Q2. What general properties of matrix $A$ would ensure Q2 be satisfied?
Another obvious class of matrix in Q2 is diagnoal matrices with positive diagonal entries.
Q2 is settled by Robert Israel($A^{-1}$ has entries all positive).