Some observations on Matrix multiplications.

To multiply a matrix of type m X n and p X q, we must have the middle numbers matching, i.e. n=p.

To multiply a matrix by itself, it must be a square matrix (m=n) to begin with.

To add two matrices together, they must be the same size, but multiplying a matrix by a number (scalar) has no size requirement.

All matrices of a fixed square size n X n can be freely multipled and added, as well as multiplied by scalars. The identity matrix denoted by or simply behaves like the number 1 for multilication. The matrix with all zero entries behaves like zero (0).

However, the product is not commutative and nonzero matrices need not have a reciprocal.

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