Given $ M = \pmatrix{a &-1 &-1\\-1 &b &-1\\-1 &-1 &c},A = \pmatrix{e\\0\\-1} $ and $a,b,c,e>0$:
What requirements should $a,b,c$ meet such that: $MX = b$ has a non negative solution (all components of $X$ are nonnegative) , given $B = \pmatrix{b_1&b_2&b_3}^T, b_1\geq 0, b_2\geq ,b_3\geq 0$ and $\max(b_1,b_2,b_3)>0$.
What requirements should $a,b,c,e$ meet such that $Mx = B$ has a solution $x = \pmatrix{x_1&x_2&x_3}^T$ given $B = \pmatrix{b_1&b_2&b_3}^T, b_1\geq 0, b_2\geq ,b_3\geq 0$ and $\max(b_1,b_2,b_3)>0$. And also satisfy that $MA = S = \pmatrix{s_1&s_2&s_3}^T$ if $x_i<0$ then $s_i <0$.