I want to prove the following:
If $X$ and $Y$ are $n \times n$ matrices, and $ X = \left[\begin{matrix} A & B\\ C & D \end{matrix}\right], Y = \left[\begin{matrix} E & F\\ G & H \end{matrix}\right] $
where A, B, C, D, E, F, G, and H are $n/2 \times n/2$ submatrices, then the product $XY$ can be expressed in terms of these blocks:
$ XY = Z = \left[\begin{matrix} AE + BG & AF + BH\\ CE + DG & CF + DH \end{matrix}\right] $
My initial thought was to use the matrix multiplication definition:
$Z_{ij} = \sum_{k=1}^n X_{ik} Y_{kj}$
and show that each $Z_{ij}$ equals the element in $Z$ by going case by case.
Case1 would be something like: $1 \leq i \leq(n/2), 1 \leq j \leq (n/2)$.
So in this case, $X_{ij} = A_{ij}$ and $Y_{ij} = E_{ij}$
I am stuck here and not sure if I am on the right track. Please advise me on how to proceed from here (or suggest an alternative method).