If I generate a random matrix what is the probability of it to be singular?

Question

Just a random question which came to my mind while watching a linear algebra lecture online. The lecturer said that MATLAB always generates non-singular matrices. I wish to know that in the space of random matrices, what percentage are singular? Is there any work related to this?

Answer 1

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.