As is well known, the Hadamard inequality is trivial from this point of view. But the Hadamard inequality is discovered so late.
When did people realize that the determinant of a matrix is actually the volume of a parallelepiped?
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1$\begingroup$ It's worth mentioning that the idea of a matrix appears much later than that of a determinant (including the determinant as a volume), so in a certain sense the question is ill-stated. $\endgroup$– Dave L RenfroMay 31, 2017 at 15:43
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According to O'Connor and Robertson, the first observation dates 250 years ago:
Lagrange, in a paper of 1773, studied identities for 3 × 3 functional determinants. However this comment is made with hindsight since Lagrange himself saw no connection between his work and that of Laplace and Vandermonde. This 1773 paper on mechanics, however, contains what we now think of as the volume interpretation of a determinant for the first time. Lagrange showed that the tetrahedron formed by $O(0,0,0)$ and the three points $M(x,y,z)$, $M'(x',y',z')$, $M''(x'',y'',z'')$ has volume $$\frac16 [z(x'y'' - y'x'') + z'(yx'' - xy'') + z''(xy' - yx')].$$
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2$\begingroup$ The Lagrange reference, not given by O'Connor and Robertson, is on page 91 of Nouvelle solution du problème du mouvement de rotation d'un corps de figure quelconque qui n'est animé par aucune force accélératrice, Nouv. Mém. Acad. Roy. Sci. Berlin 1773 (1775) 85-120. $\endgroup$ Jun 3, 2017 at 15:57