The non-square determinant is written into the laws of physics. See the attached file: Chapter 2. It is needed to find left or right inverses to non-square matrixes.
I don't understand your description/definition of the non-square determinant. Can we do a couple of examples? What is the non-square determinant of: \(\left[\begin{matrix} a & b\\ c & d\\ e & f \end{matrix}\right]\) How about \(\left[\begin{matrix} a & b & c\\ d & e & f \end{matrix}\right] \) And finally, a more complicated one: \(\left[\begin{matrix} a & b & c & d\\ e & f & g & h \\ i & j & k & l \end{matrix}\right]\) Please explain each step in the calculation.