Ahmee M. answered • 05/14/19
Berkeley Grad Student for Math, Physical Sciences, and ACT Tutoring
This will depend on the distribution you use to draw the entries. If you do something reasonable like pick the uniform distribution over [0,1] or a Gaussian over the reals, then the probability of a random matrix being singular is 0. If we think of nxn matrices as just being the elements of n^2 dimensional space, then the singular ones are just solutions to a polynomial (the determinant). In general, the zeros of a nice function form a subspace of dimension one less: in this case, n^2 - 1. Any sort of reasonable notion of size on R^{n^2} should tell us that an n^2 -1 dimensional subset has size 0.
