Proof of a conjecture concerning the Hadamard powers of inverse \(M\)-matrices (Q874999)

Let \(A\) be a real \(n\times n\) matrix. Assume that (i) all entries of \(A\) are nonnegative; and (ii) \(A^{-1}\) exists and all its off-diagonal entries are nonnegative (briefly: \(A\) is an inverse \(M\)-matrix). Let \(A^{(r)}\) denote the matrix obtained from \(A\) by raising each entry to the \(r\)th power (\(r>1\)). The author shows that \(A^{(r)}\) is also an inverse \(M\)-matrix. This verifies a conjecture of \textit{B. Wang, X. Zhang} and \textit{F. Zhang} [Linear Algebra Appl. 305, No. 1--3, 23--31 (2000; Zbl 0948.15004)].