geometric intuitions of determinant

Determinant in 2D

encodes the area under the parallelogram of with edge vectors and

Volume and Triple Product in 3D

encodes (signed) volume of parallelepiped with edge vectors :

The above relationship is known as a “triple product formula”.

Note we cannot reverse order of cross product, otherwise we get the wrong sign.

Determinant of a Linear Map

If a matrix encodes a linear map , the determinant measures the change in volume: