Probability that a random binary matrix is invertible? [duplicate]
Here's the answer over $\mathbb F_2$; I don't know about other rings:
The first row vector has a $1$ in $2^n$ chance to be linearly dependent, the second $2$ in $2^n$ and so on, so the probability for an $n\times n$ matrix to be invertible is
$$p(n)=\prod_{k=1}^{n}(1-2^{-k})\;,$$
and the limit is
$$\lim_{n\to\infty}p(n)=\prod_{k=1}^{\infty}(1-2^{-k})\approx0.288788$$
as calculated by W|A.