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What is the minimal possible rank of a square matrix, that:

  1. is $m\times m$,
  2. has elements from a finite field with $n$ elements,
  3. has $0$s on its diagonal, and
  4. has nonzero off-diagonal entries?
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  • $\begingroup$ So you're asking for the minimal rank of a matrix over a finite field? $\endgroup$ – JSchlather Jan 10 '13 at 8:34
  • $\begingroup$ Sorry, I've put enter too fast. $\endgroup$ – Jonny Jan 10 '13 at 8:35
  • $\begingroup$ Where does this problem come from? $\endgroup$ – user1551 Jan 10 '13 at 12:21
  • $\begingroup$ Don't know exactly. This is a thing for us to think after classes. $\endgroup$ – Jonny Jan 10 '13 at 14:44
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The identity matrix will have maximal rank $m$.

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  • $\begingroup$ Oh, I mean minimal of course. Sorry $\endgroup$ – Jonny Jan 10 '13 at 8:32
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    $\begingroup$ the zero matrix will have rank 0. $\endgroup$ – Ittay Weiss Jan 10 '13 at 8:33

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