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Question about an inequality described by matrices

Let $A=(a_{ij})_{1 \le i, j \le n}$ be a matrix such that $\sum_\limits{i=1}^{n} a_{ij}=1$ for every $j$, and $\sum_\limits{j=1}^n a_{ij} = 1$ for every $i$, and $a_{ij} \ge 0$. Let $$\begin{equation} …
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