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\section{Introduction}  We can draw a Cayley table under the matrix multiplication, however this is a non-commutative operation so we do not expect the table to be symmetric symmetric.  \begin{equation}  \begin{array}{ c| c c c c c c c}  & \square Let us define, $s$ as a square (all elements non-zero) matrix. $d$ and $a$ as diagonal and anti diagonal matrices respectively, then $nw,ne,sw$ and $se$ (compass directions) as triangular matrices around the respective corner.  //(Examples for explicit)  \begin{equation}  \begin{array}{ c| c c c c c c c}  & s & sw & ne & se & nw & d & a \\  \hline \\  s & s & s & s & s & s & s & s \\  sw & s & sw & s & se & s & sw & se \\  ne & s & s & ne & s & nw & ne & nw \\  se & s & s & \\  nw & s & \\  d & s & \\  a & s & \\  \end{array}