How did Binet get to his matrix multiplication algorithm and where does he speak about it?

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I would like to know how Binet got to his matrix multiplication algorithm and where (book, paper, etc) he first told the world about it.

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You might want to peruse some findings of of Oliver Knill (Harvard):

When was Matrix Multiplication Invented?

You'll find a number of potential resources to dig further. It is mentioned that Binet discussed such an algorithm in a lecture dated November 30, 1812.

From Wikipedia: Binet is "recognized as the first to describe the rule for multiplying matrices in 1812."

If it's of any solace, you might like the quote you can find on the linked webpage:

To give the history of linear algebra is a task that is as important as it is difficult.
--Nicolas Bourbaki, 1984