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Which skew-symmetric integer matrices $S$ satisfy the following inequalities $SV_i \ne z_iE_i$ for all $i = 1,\cdots, n$ where $V_i$ denotes the column with integer entries such that the $i$-th entry is $-1$ and all other entries are non-negative integers, $z_i \in \mathbb Z$, and $E_i$ denotes $i$-th standard basis vector in $\mathbb Z^n$.

The question asked originally was as follows: Notation as above can we find an integer matrix $V$ with $-1$ on the main diagonal and all other entries non-negative so that $SV$ has diagonal form.

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    $\begingroup$ for $n=2$ this is obviously not possible, and also for $n=3$ it seems excluded (irrespective of the integer constraint) $\endgroup$
    – Carlo Beenakker
    Commented Mar 26, 2020 at 12:39
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    $\begingroup$ after the last edit the question has become totally different; please, at least keep the first question visible, don't just delete it and replace it by a different question, that is really demotivating for MO users who make an effort to think about your question. $\endgroup$
    – Carlo Beenakker
    Commented Mar 26, 2020 at 13:54
  • $\begingroup$ @Carlo Beenakker Your very kind comment led us to re-think about our problem and reformulating the question in its current form is the what we really wish to ask. $\endgroup$
    – A. Gupta
    Commented Mar 26, 2020 at 15:51

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