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Has there been any work done on square matrices whose entries $a_{ij}$ are groups? (and with the operations below?)

For instance: let $\mathbb{F},\mathbb{G},\mathbb{H},\mathbb{K}$, $\mathbb{A},\mathbb{B},\mathbb{J},\mathbb{W}$ be groups, and let $\oplus$ denote group direct sum and $*$ denote free product, and for a $2\times 2$ "group matrix" define the operations$\mathbf{Y}$ and $\mathbf{U}$:

$$\left[ \begin{array}{cc} \mathbb{F} & \mathbb{G} \\ \mathbb{H} & \mathbb{K} \end{array} \right] \mathbf{Y} \left[ \begin{array}{cc} \mathbb{A} & \mathbb{B} \\ \mathbb{J} & \mathbb{W} \end{array} \right] = \left[ \begin{array}{cc} \mathbb{F}\oplus\mathbb{A} & \mathbb{G}\oplus\mathbb{B} \\ \mathbb{H}\oplus\mathbb{J} & \mathbb{K}\oplus\mathbb{W} \end{array} \right] $$

$$ \left[ \begin{array}{cc} \mathbb{F} & \mathbb{G} \\ \mathbb{H} & \mathbb{K} \end{array} \right] \mathbf{U} \left[ \begin{array}{cc} \mathbb{A} & \mathbb{B} \\ \mathbb{J} & \mathbb{W} \end{array} \right] = \left[ \begin{array}{cc} (\mathbb{F}*\mathbb{A})\oplus(\mathbb{G}*\mathbb{J}) & (\mathbb{F}*\mathbb{B})\oplus(\mathbb{G}*\mathbb{W}) \\ (\mathbb{H}*\mathbb{A})\oplus(\mathbb{K}*\mathbb{J}) & (\mathbb{H}*\mathbb{B})\oplus(\mathbb{K}*\mathbb{W}) \end{array} \right] $$

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Well, I haven't seen any work on this. Is there any particular reason you ask? – Tanner Swett Sep 25 '11 at 4:13
@TannerL.Swett: google searches involving the terms "group" and "matrices" turn up a lot of representation theory stuff, and that's not what I'm asking about. – deoxygerbe Sep 25 '11 at 4:31
@deoxygerbe: Echoing Tanner, is there a reason why you want to study such objects? Are you interested in the properties of such matrices? – Zhen Lin Sep 25 '11 at 5:18
Note that in the category of groups coproducts (which are given by free product) don't distribute over products. For example $\mathbb{Z}*(\mathbb{Z} \oplus \mathbb{Z})$ is not isomorphic to $\mathbb{Z}*\mathbb{Z} \oplus \mathbb{Z} * \mathbb{Z}$ as the ranks of their abelianizations differ. Hence, $U$ doesn't distribute over $Y,$ which makes the two operations seem algebraically unrelated. In my mind, this raises doubts as to whether someone would find such objects interesting. – jspecter Sep 25 '11 at 5:22

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