We say that $A\in M(n,n)$ is an $M$-matrix if it can be written as $A= \alpha I_n -B$, where $B\geq 0$ (inequalities taken componentwise).
I want to prove the following property of $M$-matrices:
Proposition. Let $A$ be an irreducible, non singular $n\times n$ $M$-matrix. Then it holds the Discrete Maximum Principle, i.e. if we have $y\in \mathbb{R}^n$ such that $Ay\geq 0$, then $\max\{y_1,y_n\}=\max_i\{y_i\}$
I know that an irreducible $M$-matrix $A$ is non singular and $A^{-1}>0$ iff $\alpha > \rho(B)$ (where $\rho$ is the spectral radius), iff there exists $v\in \mathbb{R}^n$ such that $v>0$ and $Av \geq 0$, $Av \neq 0$.
Any help would be really appreciated!
Edit
In the solutions of some tests about the finite difference method for discretization of pdes my professor wrote that for an M-matrix the discrete maximum principle holds (which is false, as proven in the answer below), hence the system is stable (i.e. the norm of the inverse of the matrix of the linear system is uniformly bounded from above (supremum norm)).
How can I prove that discrete maximum principle implies stability?
(If it is true; at this point I am not sure of anything!).