Some inverse \(M\)-matrix problems (Q5931894)

A nonnegative square matrix \(B\) is called a power invariant zero pattern matrix if it satisfies the condition that the \((i,j)\) entry of \(B^k\) is zero if and only if the \((i,j)\) entry of \(B\) is zero. In this paper the authors obtain an upper bound and a lower bound for \(\alpha_0\) such that \(\alpha I+B\in \mathcal M^{-1}\) for \(\alpha>\alpha_0\) and \(\alpha I+B\not\in\mathcal M^{-1}\) for \(\alpha\leq\alpha_0\), where \(B\) is a nonnegative matrix and satisfies the condition that for any positive constant \(\beta\), \(\beta I+B\) is a power invariant zero pattern matrix.