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the binary operation should also be associative (e.g. subtraction is no good)

For 2x2 matrix multiplication, it is known that Strassen's algorithm is optimal in the sense that there exists no algebraic circuit that uses only 6 "scalar" multiplications. I too…

It seemed to me that the paragraph you quoted does not much illuminate why one should choose for one or the other.

No!

But both class and className are camelCased.