# Is this block-matrix multiplication idea correct?

I am trying to get better at factoring matrices into special forms, e.g. block upper/lower triangular, so I was wondering whether I have this correct.

If matrices A through H are square and of the same size, nxn, is my matrix on the RHS correct? I basically treated the blocks as scalars, and performed the ordinary matrix multiplication techniques.

If it is correct, now what if A were mxn? Does this just require a bit more, e.g., so long as E and F are $nxm$, for the matrix multiplication to make sense, then again I can treat the blocks as scalars and proceed with ordinary matrix multiplication? (And if B were mxn, then G and H would need to be nxm, etc.)

$$\begin{bmatrix} A & B \\ C & D \\ \end{bmatrix} * \begin{bmatrix} E & F \\ G & H \\ \end{bmatrix} = \begin{bmatrix} AE+BG & AF+BH \\ CE+DG & CF+DH \\ \end{bmatrix}$$

Thanks,