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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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So you're asking for the minimal rank of a matrix over a finite field? –  JSchlather Jan 10 '13 at 8:34
    
Sorry, I've put enter too fast. –  Jonny Jan 10 '13 at 8:35
    
Where does this problem come from? –  user1551 Jan 10 '13 at 12:21
    
Don't know exactly. This is a thing for us to think after classes. –  Jonny Jan 10 '13 at 14:44

1 Answer 1

up vote 1 down vote accepted

The identity matrix will have maximal rank $m$.

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

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