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Let $A$ be a $n \times n$ matrix with entries on the set $\{0,1\}$, with exactly two ones on each column and two ones on each row.

Give necessary and sufficient conditions for rank$(A)$ to be $n$.

I found two solutions in these two articles:

http://ge.tt/1x0JYek?c

but I really don't understand them, I was wondering if there is an easier way to solve this problem, or if you could explain to me the main idea of this articles in a less advanced and easier way, and/or point out which exactly are the necessary and sufficient conditions. Thank you.

(I know a matrix $A$ is non-signular if and only if rank$(A)=n$, that's why I think these articles can help).

InsideOut
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Carlos Beris
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  • It seems the linked paper shows that if $n=4$ there are no such matrices, but for other $n$ the paper constructs one such matrix. So the "necessary and sufficient" conditions (whatever they may be) must take special exception in the $n=4$ case. – coffeemath Jun 30 '13 at 23:19
  • It is a repeat, but I don't think the answer is good. Even the answerer states that "I don't think this completely answers the question." – Calvin Lin Jul 01 '13 at 01:31
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    See my answer in another thread. The basic reason that no such matrices exist for $n=4$ is that $4$ cannot be written as a sum of positive odd integers that are larger than $1$. – user1551 Jul 01 '13 at 02:27

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