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</math>
This is the '''Cayley–Menger determinant'''. For <math>n=2</math> it is a [[symmetric polynomial]] in the <math>d_{ij} </math>'s and is thus invariant under permutation of these quantities. This fails for <math>n>2,</math>
A proof of the second equation can be found.<ref>{{Cite web|url=https://www.mathpages.com/home/kmath664/kmath664.htm|title=Simplex Volumes and the Cayley-Menger Determinant|website=www.mathpages.com|archive-url=https://web.archive.org/web/20190516033847/https://www.mathpages.com/home/kmath664/kmath664.htm|archive-date=16 May 2019|access-date=2019-06-08}}</ref> From the second equation, the first can be derived by [[Elementary row operation|elementary row and column operations]]:
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