ChapterZero

Robert Rosenbaum posted a really nice question to the UH math mailing list:

Given an matrix of integers, what is the probability that it is
invertible?

What if we put a bound on the magnitude of the integers (so that we consider only a finite number of matrices)? In other words, given an integer and an integer , what is the probability that an matrix of integers each less than in absolute value is invertible? Is there an elementary way to express this value in terms of and ? I don’t know.

If we don’t restrict ourselves to integer valued matrices and consider all real or complex valued matrices, I think that the probability would be exactly 1. Can someone give an argument for this claim?

I got the last question, where you consider all of : from an earlier problem I posted about, it is clear that the set of noninvertible matrices has empty interior, so measure zero; therefore a random matrix is invertible with probability one.

The discrete case is probably a lot harder…

Update: It seems I made an unwarranted assumption. Just because the set of noninvertible matrices has an empty interior doesn’t mean it has measure zero. I still think that’s true, but the proof is not complete.