One thing that determinants are useful for is in calculating the
area determinant of a parallelogram formed by 2 two-dimensional
vectors. The matrix made from these two vectors has a
determinant equal to the area of the parallelogram. Area
determinants are quick and easy to solve if you know how to
solve a 2x2 determinant. A parallelogram in three dimensions is
found using the cross product. One really neat application of
determinants is that they can be used to find the area of a
parallelogram. Now suppose I create a parallelogram using these
four points a, b, c and d. I've got the parallelogram graphed
here and it turns out that if you pick any vertex of the
parallelogram I'm going to pick a and look at the vectors ab and
ad that you can use those vectors to form a determinant whose
value is going to be the area. Now ab vector ab goes from point
a to point b and so its components are going to be 9-1 which is
8 -4--2 -4+2 -2 and ad goes from point a to point d and so its
components are going to be going to b