The 2 dimensional determinant of a matrix can be interpreted as the area of a parallelogram as shown in the following diagram.

This carries on through higher dimensions.  Below depicts a 3 variable system.

The rows r1, r2, r3 are vectors each. The various summations taken 1, 2 and 3 at a time define a parallelepiped. 


The following excerpt is from X and may yield some insight when maximum entropy principle is applied. ( still working on this )




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