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 $3 \times 3$ matrices, $M_1$ and $M_2$ with 
$$M_1 = \begin{pmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{pmatrix} \qquad\text{and}\qquad
M_2 = \begin{pmatrix} b_1 & b_2 & b_3 \\ b_4 & b_5 & b_6 \\ b_7 & b_8 & b_9 \end{pmatrix}$$
with the coefficients $a_n,b_n \in \mathbb{Z}$ and $1 \leq a_n,b_n \leq 9$
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 $\zeta$ 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!
 A: This answer was obtained numerical, but it is nevertheless exact
There are $N=9^9$ possibilities to choose the matrix $m$ (each with equal probability). There are 1913 possible outcomes for $\det m$. The most likely is $\det m =0$. The probability for this event is $$P[\det m = 0] = \frac{218605}{14348907} \approx 1.5\%.$$ The rest is distributed over different integer number. The largest determinant is a matrix with $\det m = 1216$. There are 3 possibilities for this matrix (therefore the probability is $P[\det m = 1216] = 3/N$). There are also 3 matrices for which $\det m=-1216$. (in fact this is always true. If there are $n$ matrices with $\det m= x$ then there are exactly $n$ matrices with $\det m=-x$). 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 $\det m$.
The absolute value of the determinant may assume 957 values. Therefore, there are $957*(957+1)/2 =458403$ pairs of matrices $m_1$ and $m_2$. Out of them there are 274487 pairs coprime. (The fraction of matrices which are coprime is $274487/458403 \approx 60\%$)
Counting each of the pair of matrices $m_1, m_2$ with the proper probability, we obtain the probability that two matrices are coprime
$$P[ m_1 \text{ and } m_2 \text{ are coprime}]= \frac{7253958722902984}{16677181699666569} \approx 43\%.$$
A: This paper actually shows that the probability two large matrices with integer entries are co-prime is $\frac{1}{3}$, which actually is a bigger version of your question.


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*http://www.mccurley.org/papers/relprime.pdf
