Probability that a random binary matrix is invertible - what is going on?

Question

This is a follow-up to this question: Probability that a random binary matrix is invertible?

The answer says that the probability of a random $\{0,1\}$, $n \times n$ matrix to be invertible is:

$$p(n)=\prod_{k=1}^{n}(1-2^{-k})\;,$$

For a $32\times32$, that's about 0.288.

But, when I generate a random matrix in Matlab, and check its rank, it's always 32! The code is: A=randi([0 1],32,32);rank(A). You can even try it online here.

Is the answer wrong? Is Matlab/Octave wrong? Please help me solve the mystery. Thanks!

Answer 1

That is a special result, for algebra modulo 2, binary. That is what $\mathbb{F}^2$ means.

For any of the following sets of random matrices:

$$ A \in \mathbb{R}^n \times \mathbb{R}^n,\{a_{ij} \sim N(0,1)\}\\ A \in \mathbb{R}^n \times \mathbb{R}^n,\{a_{ij} \sim U[0,1]\}\\ $$

Through any of the following standard commands, respectively:

rank(randn(32,32))
rank(rand(32,32))

The matrix has full rank almost surely, being the probability of having a full rank matrix equal to 1, as the standard interpretation of probability under infinite sets, there are possible values, but under a measure not comparable with the whole set measure.

By other side, the following set of random matrices: $$ A \in \mathbb{R}^n \times \mathbb{R}^n,\{a_{ij} \sim U[0,1]\}\\ $$

Realizable throught the command:

rank(randi([0,1],32,32))

The probability of having a full rank matrix is at least $1-(3/4+\sigma(1))^n$.

https://arxiv.org/abs/math/0501313

Probability that a random binary matrix is invertible?

If I generate a random matrix what is the probability of it to be singular?

https://mathoverflow.net/questions/12657/proving-almost-all-matrices-over-c-are-diagonalizable

https://en.wikipedia.org/wiki/Almost_surely