applying M to them:(Mv) · (Mw) = v · w.[51]Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group.[52] General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory.[53]අපරිමිත න්‍යාස[සංස්කරණය]It is also possible to consider matrices with infinitely many rows and/or columns[54] even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.If R is any ring with