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

A: See

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*J. Binet, Mémoire sur un Système de Formules analytiques, et leur application à des considérations géométriques, Journal de l’École Polytechnique 9 (1813), 280–354.

Binet gave this lecture on 30 November 1812, the same day Cauchy presented on the same result (now known as the Cauchy–Binet formula); see

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*A. L. Cauchy, Mémoire sur les Fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu’elles renferment, Journal de l’École Polytechnique 10 (1815), 29–112.

