Probability that a random binary matrix is invertible

The two latest posts have involved invertible matrices with 0 and 1 entries. If you fill an n × n matrix with 0s and 1s randomly, how likely is it to be invertible? What kind of inverse? There are a couple ways to find the probability that a binary matrix is invertible, depending on what you mean by the inverse. Suppose you have a matrix M filled with 0s and 1s and you’re looking for a matrix N such that MN is the identity matrix. Do you want the entries of N to also be 0s and 1s? And when you multiply the matrices, are you doing ordinary integer arithmetic or are you working mod 2? In the previous posts we were working over GF(2), the field with two elements, 0 and 1. All the elements of a matrix are either 0 or 1, and arithmetic is carried out mod 2. In that context there’s a nice expression for the probability a square matrix is invertible. If you’re working over the real numbers, the probability of binary matrix being invertible is higher. One way to see this is that the inverse…