What would be the probability of matrix having determinant zero out of all matrices with all entries being positive? How does one calculate such?

Edit: Restriction to natural numbers and size of $n \times n$. Restriction to entries from 0 to 5 or from 0 to 10.

  • 5
    $\begingroup$ How are the entries chosen? If from the reals with independent uniforms on $[0,1]$ say, the probability is $0$. Indeed the answer is $0$ for any independent random variables with any continuous distribution. $\endgroup$ Dec 10, 2012 at 7:30
  • $\begingroup$ You can't answer this question unless you know what the distribution is. $\endgroup$
    – Joe Z.
    Dec 10, 2012 at 7:30
  • $\begingroup$ Even with your edit, you still can't answer it unless you know the probability that each number occurs. $\endgroup$
    – Joe Z.
    Dec 10, 2012 at 7:36
  • $\begingroup$ After my edit, is this answerable? Oh.. but then distribution... OK, I will provide distribution. $\endgroup$
    – DDR
    Dec 10, 2012 at 7:39
  • $\begingroup$ The distribution(s) you chose seem(s) to eliminate all hope of an underlying structure to the problem. $\endgroup$
    – Did
    Dec 10, 2012 at 7:51

1 Answer 1


If the possible elements belong to an infinite set such as natural numbers, real numbers or complex numbers, then the probability is 0 simply because the space of matrices with determinant 0 is a subspace of all matrices.

If they belong to a finite set, then the probability is between 0 and 1 but most likely so small and gets even smaller as your set grows.


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