Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible?
Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$.
Is there an explicit formula as a function of $n$? Does it tend to 1 as $n$ grows large?
I’m sure this is all known…


Solutions Collecting From Web of "Probability that a random binary matrix is invertible?"

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


and the limit is


as calculated by W|A.