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!