system
Given the following post, respond as if you're commenting on Reddit:
human
What's the probability that in n vectors, each with n elements where each element is a random integer, that the set is linearly independent?
gpt
You are asking the question:  "Given an N x N matrix *A* with random entries from some commutative ring *R* (e.g., integers), what is the probability that *A* has non-zero determinant?" The word "random" in this particular context means that we have assigned some probability distribution to the ring *R* and the matrix entries are chosen according to that distribution. (In particular, there is no uniform distribution on the integers, so your question doesn't make sense as stated since an unqualified "random" is usually taken to mean "uniform".)

This question is pretty tough and there are entire branches of mathematics that describe random matrixes. One easy-ish result is the following. Suppose the ring *R* is just the real numbers. The condition that the matrix *A* has zero determinant is equivalent to some equation of the form p(a*_ij_*) = 0, where *p* is some polynomial and the a*_ij_* are the entries of the matrix. This equation describes some set in R^(N^2). All non-zero polynomials on R^(m) are non-zero (Lebesgue) almost everywhere, whence the set of points that satisfies p(a*_ij_*) = 0 has Lebesgue measure zero. Now suppose we give the real numbers *any* probability measure. That measure is absolutely continuous with respect to Lebesgue measure, and so the zero set of *p* has zero probability. In other words, the probability that a random matrix is *singular* is zero, no matter what the probability distribution on the reals is.

For other rings, the answer may be different. For instance, if *R* = **Z***_2_* and we give *R* the uniform distribution, the probability that *A* is singular is actually positive for all *N*. (In fact, the probability remains positive as *N* --> infinity.)