Probability of determinants being coprimeI have a question that is not of particular significance, but I would love to understand the underlying principles.Suppose we have two square 3 × 3 matrices, M 1 and M 2 with M 1 = ( a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 ) and M 2 = ( b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 ) with the coefficients a n , b n ∈ Z and 1 ≤ a n , b n ≤ 9What 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 ζ 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!

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Probability of determinants being coprimeI have a question that is not of particular significance, but I would love to understand the underlying principles.Suppose we have two square   3  ×  3 matrices,       M    1   and       M    2   with      M    1    =      (                                        a            1                                                a            2                                                a            3                                                            a            4                                                a            5                                                a            6                                                            a            7                                                a            8                                                a            9                                )      and        M    2    =      (                                        b            1                                                b            2                                                b            3                                                            b            4                                                b            5                                                b            6                                                            b            7                                                b            8                                                b            9                                )  with the coefficients       a    n    ,      b    n    ∈      Z   and   1  ≤      a    n    ,      b    n    ≤  9What is the probability that the matrices&#039; determinants are coprime, when uniformly random coefficients satisfying the conditions are chosen.I am familiar with the Riemann&#039;s   ζ 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!

Jaylah Washington · Accepted Answer

This answer was obtained numerical, but it is nevertheless exactThere are   N  =      9    9   possibilities to choose the matrix m (each with equal probability). There are 1913 possible outcomes for detm. The most likely is detm=0. The probability for this event is  P  [  det  m  =  0  ]  =      218605    14348907    ≈  1.5  %  .The rest is distributed over different integer number. The largest determinant is a matrix with detm=1216. There are 3 possibilities for this matrix (therefore the probability is   P  [  det  m  =  1216  ]  =  3      /    N. There are also 3 matrices for which detm=−1216. (in fact this is always true. If there are n matrices with detm=x then there are exactly n matrices with detm=−x). To find out if the determinants are coprime the sign does not matter, therefore we only need to know the probabilities  P  [      |    det  m      |    =  x  ]  .Generically, the probability decreases for increasing detm.The absolute value of the determinant may assume 957 values. Therefore, there are 957∗(957+1)/2=458403 pairs of matrices m1 and m2. Out of them there are 274487 pairs coprime. (The fraction of matrices which are coprime is   274487      /    458403  ≈  60  %Counting each of the pair of matrices       m    1    ,      m    2   with the proper probability, we obtain the probability that two matrices are coprime  P  [      m    1     and       m    2     are coprime  ]  =      7253958722902984    16677181699666569    ≈  43  %  .

Probability of determinants being coprime I have a question that is not of particular significance,

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