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Consider the following system of linear equations over $x_{ij}$ for $1\leq i\leq m$ and $1\leq j\leq n$:

$\sum_{j}x_{ij}=a_i$ for $i=1, \cdots, m$ and $\sum_{i}x_{ij}=b_j$ for $j=1, \cdots, n$ where each $a_i$ or $b_j$ is positive and $\sum_i a_i= \sum_j b_j$.

Is it the case that there always exists a positive solution, namely, for each $i,j$, $x_{ij}>0$. If this is true, is there any simple proof?

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up vote 3 down vote accepted

Denote $S=\sum_{i=1}^m a_i=\sum_{j=1}^n b_j$. Then $x_{ij}=\frac{a_ib_j}{S}$ is a positive solution.

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