>>11227592Just apply Cavalieri’s principle to parallelepipeds
Hence, the elementary row operations used to factorize the determinant also effect the volume in analogous ways. Eliminating the elements in a row has no effect, as that elementary matrix has determinant 1, and displacing a vector of the parallelepiped by vectors in the base does not change the metric of the base, or the altitude of the vector. Swapping vectors can only change the sign of the determinant, not the magnitude, so, if your parallelepiped has no concept of signed volume, then the absolute value of the determinant, and the volume of the parallelepiped will effected the same. Lastly, scaling a vector also scales the height, so, the absolute value of the determinant, and the volume of the parallelepiped will be effected the same. Lastly, if you take the volume of the unit parallelepiped to be 1, the same as the determinant of the identity matrix, you will get the same values for the absolute value of the determinant, and the volume of the parallelepiped.
