Remark

The above formulae extend to a more general situation. To be precise:

(a) Suppose $H = \bigcup_{(\iota, \kappa) \in I \times K} G_{\iota \kappa}$ where each $G_{\iota \kappa}$ is a commutative group written additively; then the sum $M + M'$ may be defined when, for each ordered pair $(\iota, \kappa)$, $m_{\iota \kappa} \in G_{\iota \kappa}$ and $m'_{\iota \kappa} \in G_{\iota \kappa}$.

(b) Let I, K, L be three sets with K assumed finite and let $H' = \bigcup_{(i, k) \in I \times K} H'_{ik}$, $H'' = \bigcup_{(k, l) \in K \times L} H''_{kl}$, $H = \bigcup_{(i, l) \in I \times L} H_{il}$ be three sets; suppose that each $H_{il}$ is a commutative group written additively and for each triple $(i, k, l)$ let

be a mapping. Then if $M' = (m'_{ik})_{(i, k) \in I \times K}$, $M'' = (m''_{kl})_{(k, l) \in K \times L}$ are matrices such that $m'_{ik} \in H'_{ik}$ and $m''_{kl} \in H''_{kl}$ for all $i, k, l$ we can define the product $M'M''$ via the $f_{ikl}$. We leave to the reader the task of writing down and proving the formulae analogous to (4), (5) and (6).