What sorts of questions are you trying to answer?

Obviously you can only hope to say something about the *distribution* of solutions Ax = 0 is A if a random matrix.

But if you are simply given a linear system, there are myriad effective numerical techniques to efficiently compute solutions. So are you just looking for efficient numerical techniques given that A has such a structure?

Or is it something deeper and statistical you want to say about the solutions?

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>In other words, in our overdetermined system we essentially have sets of duplicate equations as the coefficients come from the same distribution.

*Duplicate* isn't that meaningful of a word in this context. Perhaps a system where an equation is *linearly dependent* on the other equations is more precise. But just because you're generating coefficients from the same distribution doesn't mean they equations will be linearly dependent.

For example if the coefficient distribution is just a simple binomial {0, 1}, then we can obviously get systems of equations like x = 0 and y = 0, which has a unique solution, or 0 = 0, which is underdetermined and has infinitely many solutions.