Equivalence of pairs of matrices with relatively prime determinants over quadratic rings of principal ideals

A special equivalence of matrices and their pairs over quadratic rings is investigated. It is established that for the pair of $n\times n$ matrices $A,B$ over the quadratic rings of principal ideals $\mathbb Z[\sqrt k]$, where $(\operatorname{det}A,\operatorname{det}B)=1$, there exist invertible matrices $U\in GL(n,\mathbb Z)$ and $V^A,V^B\in GL(n,\mathbb Z[\sqrt k])$ such that $UAV^A=T^A$ and $UBV^B=T^B$ are the lower triangular matrices with invariant factors on the main diagonals.