Determinant

Properties of the Determinant

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