# The terminologies "Adjoint" and "Adjugate"

The adjoint of a matrix is also called the adjugate.

However, the word "adjoint" is also used in Linear Algebra for linear transformations, and the "adjoint of a matrix" and "adjoint of a linear map" are very different notions.

Question 1: For matrices, which word was used first - "adjoint" or "adjugate" (can one give a historical reference)?

Question 2: Can one give the historical reference for the identity $$A\, \mathrm{adj}(A)=\mathrm{det}(A)I$$

• Cramer's rule is the origin of the equality of the second question. At that time (1750 by Cramer and apparently even earlier by Leibniz) there were no determinants. Until Cauchy came. A quick look into Cauchy's Sur les fonctions qui ne peuvent obtenir que deux valeurs suggests that he called it adjoint et conjugué (p.135 of the Oeuvres complètes d'Augustin Cauchy. Série 2, tome 1), but someone should read backwards whether these terms are defined somewhere earlier. It doesn't seem to be the first appearance in Cauchy's work.
– Ben
Jul 29, 2015 at 12:40
• It surprises me that "adjoint" and "adjugate" are taken as synonyms by anyone. The use of "adjoint" for matrices and/or linear transformations are slightly different presentations of the same thing, so far as I know, certainly not different in any substantive way. Jul 30, 2015 at 19:57
• Dear @paulgarrett, I was about to write the same thing, but then I realised that actually some people do (did?) refer to the adjugate as the adjoint. Unfortunately, I didn't find the time to read more in Cauchy to be sure what he called it at first or later on. I was very surprised when I found this in french writing, since nowadays, most french mathematicians refer to is as 'matrice complémentaire'. By the way, the french Wikipedia also says "Dans la littérature anglo-saxonne, la matrice complémentaire (transposée de la comatrice) est parfois appelée «matrice adjointe»[...]"
– Ben
Jul 31, 2015 at 10:38
• @paulgarrett, I'm not surprised the terms could be synonyms. I first learned the term adjoint for matrices as meaning "the inverse matrix but without dividing by the determinant" and later I learned the other meaning of adjoint that is connected with inner products (or more generally symmetric bilinear forms). The fact that the matrix having the first meaning is also called the "classical adjoint" suggests that the term adjoint really has had these two very different senses in linear algebra, with "adjugate" probably being someone's attempt to make a new term for the first concept of adjoint.
– KCd
Aug 3, 2015 at 19:07
• @KCd, hm!! Thanks for the extra data point. :) Given the number of archaic books I've read, I'm surprised I'd not encountered this use before... Aug 3, 2015 at 19:19

## 2 Answers

According to the Oxford English Dictionary, the first use of the word 'adjugate' is by CL Dodgson (Lewis Carroll) Proc. Roy Soc 15(1866) at p. 151. He says "In doing so I shall use the word "adjugate" in the following sense:-if there be a square block, and is a new block be formed, such that each of its terms is the determinant of the complemental minor of the corresponding term of the first block, the second block is said to be adjugate to the first" (italics in original). I take his use of inverted commas to be an admission that he is defining a new word.

In Larson/Falvo sixth ed. (2009) the adjoint is defined as the transpose of the cofactor matrix. The term "adjugate" is not used and does not show up in the index. So apparently some continue to use the term adjoint to mean classical adjoint or adjugate.