# Minimal rank of a matrix with zero diagonal and nonzero off-diagonal entries over a finite field

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

The identity matrix will have maximal rank $m$.