Too long for a comment, but this premise of the question:
I know that matrix multiplication was introduced by Cayley (correct me if I am wrong)
is indeed wrong. Gauss in Disquisitiones Arithmeticae (1801) has something called not matrix multiplication but combination of substitutions — e.g. he writes near the end of §294:
$(S)=\left\{\!\!\begin{smallmatrix} \phantom{-}7245,&\phantom{0}5,&\phantom{-}22\\ -2415,&\phantom{0}2,&-28\\ 19320,&25,&\phantom{0}-7 \end{smallmatrix}\!\right\}$ combined with $(S')=\left\{\!\!\begin{smallmatrix} \phantom{-000}3,&\phantom{-000}5,&\phantom{-00}1\\ -2440,&-4066,&-813\\ \phantom{0}-433,&\phantom{0}-722,&-144 \end{smallmatrix}\!\right\}$ produces $\left\{\!\!\begin{smallmatrix} \phantom{-}9,&11,&12\\ -1,&\phantom{0}9,&-9\\ -9,&\phantom{0}4,&\phantom{0}3\end{smallmatrix}\!\right\}$.
His definition is in §270:
