Determinants

A determinant looks like nothing more than a number computed from a square matrix, but what it really does is let you see at a glance whether a transformation “flattens” space. In two dimensions the determinant is the scaling factor of area; in three dimensions it is the scaling factor of volume. A determinant of zero means the transformation squashes some direction down to zero — one dimension of space has collapsed.

This chapter starts from the definition of the determinant and explains why it is a scalar rather than a matrix; it then reveals its geometric meaning through visualisation, so that you can see what the same number signifies in two and in three dimensions. Next it gives several methods that can actually be used to compute it, so that you are not left able only to recite the definition; and it explains why “the determinant is nonzero” and “the matrix is invertible” are one and the same statement. It finally lands on concrete applications such as Cramer’s rule and the computation of areas and volumes.

The first two chapters established the viewpoint of “transformations”; this chapter begins with the first ruler for judging the nature of a transformation.