Almost all random matrices are nonsingular

tl;dr

1. Here I present an elementary proof for a classical result in random matrix theory that applies to any random matrix sampled from a continuous distribution.

One of its many important consequences is that almost all linear models with square Jacobian matrices are invertible.

2. This is also relevant to scientists that want stable internal models for deep neural networks since a deep network is an exponentially large ensemble of linear models with compact support.