Why was matrix multiplication defined the way it is?

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SUMMARY

The definition of matrix multiplication is rooted in the need to represent the composition of two linear transformations, A and B, as a single matrix C. This operation allows for the sequential application of transformations, which is essential in linear algebra. The historical motivation stems from the realization that combining these transformations can be effectively captured through the matrix multiplication operation, thereby simplifying calculations and theoretical understanding.

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quasar987
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What was the (historical) motivation for defining the rules of matrix multiplication the way it is?
 
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As far as I know, it comes from performing 2 linear transformations, A and B, one after the other. Then realising that both linear transformations can be written as one matrix, C, and then realising that C can be written as the A and B if we define this new operation called matrix multiplication.
 

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