Re: [math-fun] Zero matrix products question

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