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The Cauchy-Binet formula in linear algebra gives the determinant of a square matrix $AB$ for rectangular matrices $A,B$ of appropriate shape in terms of the determinants of the submatrices of $A,B$. (See the linked Wikipedia page for the specifics.) In the special case of $A$ with two columns, this reduces to a summation identity which Wikipedia in turn refers to as the Binet-Cauchy identity.

Why Cauchy-Binet for the determinant formula and Binet-Cauchy for the special case?

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    $\begingroup$ Who knows the mind of Wikipedia? :) Shafarevich-Remizov (p.69) refer to the identity as still Cauchy-Binet, and Dym (p.103), Lancaster (p.39), etc., refer to the formula as Binet-Cauchy, which is in line with the alphabetical convention in mathematics but is not always followed. $\endgroup$
    – Conifold
    Commented Mar 7, 2017 at 21:24
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    $\begingroup$ One would expect to alphabetize the names only if it comes from a joint paper by the two authors Binet and Cauchy. For principles/theorems with two mathematicians contributing separately, more common would be chronological order: Cauchy-Buniakovsky-Schwarz inequality; Heine-Borel theorem, etc. $\endgroup$
    – Gerald Edgar
    Commented Mar 8, 2017 at 1:01

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I am not sure what the editors of Wikipedia had in mind when arranging the names (if anything). Linear algebra textbook authors have them arranged every which way for both the formula and the identity, see e.g. Shafarevich-Remizov (p.69), Dym (p.103) and Lancaster (p.39). The formula was discovered independently but almost simultaneously in 1812 by both Binet and Cauchy.

Oliver Knill reports that, according to Sternberg, Binet not only derived the formula but was also first to derive the rule for multiplying matrices about the same time, and presented it in a lecture on November 30, 1812 (the usual attribution is to Cayley's 1857 papers). Unfortunately, this is unconfirmed:

"My own digging on Binet did not get far. It would be especially interesting to get a hand on the publication or talk of Binet of 1812 and compare it with Cauchy who would prove at the same time the Cauchy-Binet formula. Cauchy and Binet lived at around the same time, went to the same school, competed for the same jobs. Its not surprising that they discovered the Cauchy-Binet formula at around the same time. They might even have communicated about it (even so this is only speculation).

[...] It is interesting to me that determinants have appeared before matrix algebra or even matrices and that the multiplication rule for determinants predates the discovery of matrix multiplication. But in this case one can understand the reason: Cauchy-Binet is useful when trying to understand solutions of linear equations. The later can be understood also without matrix algebra, as it happened historically."

Knill's webpage also has references, timeline of the early history of determinants, and links to the current research on generalizations of the formula.

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For a discussion of Cauchy's contribution, see pages 92-131 of T. Muir, The Theory of Determinants in the Historical Order of Development, vol. 1, second edition, MacMillan, London, 1906.

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