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Some time ago, I published the following question Geometric meaning of the determinant of a matrix on the geometric meaning of determinant.

Usually, on the books of algebra, the determinant is presented as an application from the space of the matrices to a field (set of real numbers, for example) that meets a list of properties. I have read in some magazines, that this definition is due to the Bourbaki group while the geometric meaning of the determinant would be the first historical approach to this concept. Today I would like to ask for confirmation of this to those who know more than me. Thank you very much.

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Mark
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Reading some of the historical sources, it is quite clear that the geometric interpretation of determinants is not historically the first one to appear. Indeed determinants started out (before the nineteenth century) as a particular algebraic expression in terms of a collection of coefficients (but not yet arranged into a matrix considered as a single object), in a way similar to the discriminant of a polynomial or resultants. Determinants then served to determine whether a linear system of equations had a unique solution or not, whence the name. Only later do determinants appear as associated to matrices (names of Sylvester, Cayley, Hamilton seem associated to this development) considered themselves as single values. The geometric interpretation would seem to first appear with the Jacobian determinant.

The preference of a characterisation of the determinant (as a multi-linear alternating form with value $1$ on the identity matrix) over a straightforward definition (Leibniz formula) is certainly more recent still, quite possibly due to the influence of Bourbaki.

Marc van Leeuwen
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