Solution
= Solution
Put $G=M^{-1}N\ge0$ and note $A=M(I-G)$. If $\rho(G)<1$, then $A^{-1}=(I-G)^{-1}M^{-1}\ge0$. Conversely, if $A^{-1}\ge0$, take $x\ge0$, $x\ne0$, with $Gx=\rho x$. Since $M^{-1}=A^{-1}(I-G)$, $M^{-1}x=(1-\rho)A^{-1}x$. The left side is nonnegative and nonzero, forcing $1-\rho>0$. Finally the iteration error is $e_{k+1}=Ge_k$, so part (a) gives convergence for every start exactly under this condition.