What is the probability that a symmetric $3 \times 3$ invertible matrix has all integer eigenvalues? What if we limit the integer entries to have absolute values less than 10?

The probability of a random integer matrix having integer eigenvalues is small:


So far, I've worked with two exceptional cases where the matrix had all integer eigenvalues. But, most of the time, the eigenvalues are irrational numbers.

  • $\begingroup$ With what probability distribution are you selecting your matrix? $\endgroup$ – Omnomnomnom Sep 21 '16 at 2:20
  • $\begingroup$ I suppose uniform...? $\endgroup$ – Daniel Lautzenheiser Sep 21 '16 at 4:20

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