determinants

I need to revise my previous post about determinants.

The determinant is n!*(volume of simplex), as the determinant in fact measures the area of the parallelpiped delineated by the vertices (a_1, ... a_n).  To get the volume of the simplex one must divide the parallelpiped by the number of simplexes (simplicies?) contained (n!).

For example, in R^2 (two-dimensional space) for a 2x2 determinant, det [a, b; c, d]=ad-bc. This gives the volume (area) of a parallelogram.  This can be divided into n!=2!=2 simplexes (triangles).

Therefore the signed (oriented) volume of a simplex is the determinant divided by n!.