# Is Matrix Direct Sum Distributive Over Matrix Multiplication?

Follow up to this question I asked here: Is Matrix Direct Sum Distributive over Matrix Addition?. I am now wondering whether it is distributive over matrix multiplication as well? By that I mean whether the following statement is true or not:

$\bigoplus_{i=1}^n A_iB_i = \left(\bigoplus_{i=1}^n A_i\right)\left(\bigoplus_{i=1}^n B_i\right)$

By the argument given in the previous question I would take that the statement is true as in each subspace $i$ we simply have $A_iB_i$ and whether you take the matrix multiplication of the direct sum first should not matter as either way we won't have $A_iB_j$ for $i\neq j$. Can anyone confirm this result/my reasoning? Thank you!

I believe yes - taking the $2\times2$ case we get: