A contribution to the theory of M-matrices (Q910462)

Recall that a real \(n\times n\) matrix \(A=(a_{ij})\) with \(a_{ij}\leq 0\) for all \(i\neq j\) is an M-matrix if A is nonsingular and all the elements of \(A^{-1}\) are nonnegative. The author proves that if \(A=(a_{ij})\) is an \(n\times n\) M-matrix with \(a_{ii}>0\) for \(i=1,2,...,n\) then \[ \sum_{1\leq i<j\leq n}\frac{a_{ij}a_{ji}}{a_{ii}a_{jj}}<\frac{n- 1}{2}\quad for\quad n=3,5,7,... \] \[ \sum_{1\leq i<j\leq n}\frac{a_{ij}a_{ji}}{a_{ii}a_{jj}}<\frac{n}{2}\quad for\quad n=2,4,6,...\quad. \]