Let $A$ be an $n\times n$ matrix, with entries only from $\{1,\dots ,n\}$, with no repeated values in any row or any column.
Consider the system $AX=B$ of $n$ linear (Diophantine) equations $AX=B$, where $B$ is a vector with positive entries.
We are given that there is a nonnegative integer solution for the system, say $X_0$. That is, $AX_0=B$ and each entry of $X_0$ is nonnegative.
My question is the following.
First, for $\sigma \in S_n$, let $B_\sigma$ denote the vector whose entires are a permutation of the entries of $B$ using $\sigma$. Is there a permutation $\sigma\in S_n$ such that the system $AX=B_\sigma$ has a positive (not necessarily integer) solution that is not just a permutation of the entries of $X_0$?
Motivation: For specific examples of $A$ and $B$ (when $n=5$ and $n=6$), I actually found, using Sage, that for any permutation $\sigma$, the system $AX=B_{\sigma}$ has a positive solution. But I can't quit see why this is the case! One of my examples: $$A=\left[\begin{array}{ccccc}1&2&3&4&5\\2&4&5&3&1\\3&5&2&1&4\\4&3&1&5&2\\5&1&4&2&3\end{array}\right],$$ $$B=\left[\begin{array}{c}10\\11\\12\\13\\14\end{array}\right].$$ Then $AX=B_\sigma$ has a positive solution, for any $\sigma \in S_5$.