On equivalence of pairs of matrices, which determinants are primes powers, over quadratic Euclidean rings

Abstract

We establish that a pair of matrices, which determinants are primes powers, can be reduced over quadratic Euclidean ring $\mathbb{K}=\mathbb{Z}[\sqrt{k}]$ to their triangular forms with invariant factors on a main diagonal by using the common transformation of rows over a ring of rational integers $\mathbb{Z}$ and separate transformations of columns over a quadratic ring $\mathbb{K}$.