Systems of Differential Equations · Superchallenge problem

How many $2\times2$ matrices $A$ with every entry in $\{-2,-1,0,1,2\}$ have both of these properties: (i) every solution of $\mathbf x'=A\mathbf x$ that starts in the closed first quadrant stays in it for all $t\ge0$; (ii) every solution tends to $\mathbf 0$ as $t\to\infty$?