Can entries of an matrix be matrices?

Question

I wonder that can entries of an matrix be matrices?

If it has such a possibility, how do we multiply it with another matrix whose entries are real numbers?

Answer 1

You can see entries of matrices as elements of a ring (instead of a field, as we usually do with real or complex matrices). In fact, in order to get usual operations, you need addition and product of elements of matrices with some sane hypotheses on these operations (commutativity, associativity, distribution laws,...), and rings work well for that matter.

And, as matrices over fields form a ring, you can safely have spaces of matrices $M_k(M_n(\Bbb R))$ - "matrices of size $k$ whose elements are real matrices of size $n$".