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 )