Given $B>0$ and $n\in\Bbb N$ what is the probability that a given $n\times n$ integer matrix with all entries bound by absolute value $<B$ is non-singular? I am looking for precise scaling.



On non-singularity of integer matrices with bounded entries
If I understand the question correctly, this is a known hard problem if $B$ is fixed, but $n$ is growing. See, for example, this paper, written by some fairly smart people:


On the singularity probability of discrete random matrices
Jean Bourgain, Van Vu, P. M. Wood (sorry, insert citation is failing), JFA 2010.


However, if $n$ is fixed, but $B$ is growing, this is well-understood, see the paper by Yonatan Katznelson (DMJ, 1994, "Singular matrices and a uniform bound for congruence subgroups of $SL(n, \mathbb{Z})$"), his [main result](https://www.dropbox.com/s/dcsefnj5b8egjcq/Screenshot%202017-03-04%2012.35.38.png?dl=0) is that the probability is of order $\log B/B^n,$ with explicit constants.


