Given an $n\times s$ matrix $P$ of positive real numbers and $T\geq n$, find (either by a formula or an algorithm) the real symmetric $n\times n$ [Z-matrix](http://en.wikipedia.org/wiki/Z-matrix_(mathematics)) $A$ which maximizes $\min\limits_{1\leq i<j\leq n}|Q_i-Q_j|_2$ subject to 
$$
(1,1,\dots,1)A=(1,1,\dots,1),\quad tr(A)=T
$$
where $Q_i$ is the $i$th row of $A^{-1}P$ and $|\cdot|_2$ is the Euclidean norm. 

_Remark:_ Any real symmetric Z-matrix $A$ satisfying $(1,1,\dots,1)A=(1,1,\dots,1)$ is an [M-matrix](http://en.wikipedia.org/wiki/M-matrix), so has positive inverse.