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] $