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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…

Thanks!

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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.

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