Nice simple example -- oh, well, I guess that was a non-question, sorry. Okay, here are two cases where there are only finitely many matrices, hence finitely many in the answers: 1) Consider NxN matrices with entries in Z/2Z. 2) Consider NxN matrices with entries in Z, each equal to 0 or 1. --Dan On Aug 21, 2014, at 1:33 AM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 21/08/2014 02:12, Dan Asimov wrote:
Given N in Z+, what is the largest possible size f(N) of a set X of NxN matrices over Z such that
1) Any pair of them multiply to the zero matrix;
2) Each member of X has no common factor among all N^2 of its entries.
???
Having spent only a few minutes on this, it seems clear that f(N) >= 1 + floor(N/2)^2 (exercise).
Maybe it's obvious, but I don't even see why f(N) must be finite (though I'd guess it is).
Any two matrices of the form
[0 1 0] [0 0 0] [0 a 0]
have zero product and no common factor among all entries.
-- g
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