Rings and groups of matrices with a nonstandard product

We define a new operation of multiplication on the set of square matrices. We determine when this multiplication is associative and when the set of matrices with this multiplication and the ordinary addition of matrices constitutes a ring. Furthermore, we determine when the nonstandard product admits the identity element and which elements are invertible. We study the relation between the nonstandard product and the affine transformations of a vector space. Using these results, we prove that the Mikhaĭlichenko group, which is a group of matrices with the nonstandard product, is isomorphic to a subgroup of matrices of a greater size with the ordinary product.