For example, the determinant of a transformation would be 33 if that transformation increases the area of a region by a factor of 3.
The determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point, since the area of every region would then become 0.
That last one is especially important; checking if the determinant of a given matrix is 00 will give a way of computing whether or not the transformation associated with a matrix squishes everything into a smaller dimension.
Determinant for 3D matrics affect volume instead of area
