Counting integer matrices with square-free determinants

We consider the set ${\mathcal{M}}_n(\mathbb{Z};H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain and asymptotic formula on the number of matrices from ${\mathcal{M}}_n(\mathbb{Z};H)$ with square-free determinants. We also use our approach with some further enhancements, to obtain an asymptotic formula for the sums of the Euler function with determinants of matrices from ${\mathcal{M}}_n(\mathbb{Z};H)$.