Probability of determinants being coprime
I have a question that is not of particular significance, but I would love to understand the underlying principles.
Suppose we have two square matrices, and with
with the coefficients and What is the probability that the matrices' determinants are coprime, when uniformly random coefficients satisfying the conditions are chosen.
I am familiar with the Riemann's function way to find out the probability of two random integers being coprime, but I have no clue how to apply that here with additional conditions on the numbers given.
I did test it mechanically, using Mathematica and the result is around 30%, but I would like to see a proper way to do it.
I would love to at least get a few pointers as what to research to tackle this problem.
Thank you very much!