The domain of an matrix is the set of all matrices, for all positive integers . The only requirement is that the matrix multiplication be defined. The condition I've listed is the requirement necessary.
Well, there's not a whole lot to elaborate on. You're considering the linear operator consisting of matrix multiplication on the left (that is, the operator matrix is on the left of its operand and multiplies it using regular matrix multiplication). The natural domain of a function or operator is the set of all things that you can "plug in" to the operator. For these operators, the domain is what I've described above.
Is there some more specific portion of this for which you'd like more info?