In this chapter we introduce one of the most important computational tools in linear algebra – the determinants. First we discuss some motivational examples. Next we present the definition and basic properties of determinants. Then we study some applications of determinants.

Motivational examples

We now present some examples occurring in geometry, algebra, and topology that use determinants as a natural underlying computational tool.

3.1.1 Area and volume

Let u = (a1, a2) and υ = (b1, b2) be nonzero vectors in ℝ2. We consider the area of the parallelogram formed from using these two vectors as adjacent edges. First we may express u in polar coordinates as

u = (a1, a2) = ∥u∥ (cos θ, sin θ).

Thus, we may easily resolve the vector υ along the direction of u and the direction perpendicular to u as follows

Here c2 may be interpreted as the length of the vector in the resolution that is taken to be perpendicular to u. Hence, from (3.1.2), we can read off the result

c2 = ±(b2 cos θ − b1 sin θ) = |b2 cos θ − b1 sin θ|.

Therefore, using (3.1.3) and then (3.1.1), we see that the area σ of the parallel- ogram under consideration is given by

σ = c2 ∥u∥ = |∥u∥ cos θ b2 − ∥u∥ sin θ b1 | = |a1b2 − a2b1|.

Thus we see that the quantity a1b2 − a2b1 formed from the vectors (a1, a2) and (b1, b2) stands out, that will be called the determinant of the matrix

written as det (A) or denoted by

Since det(A) = ±σ, it is also referred to as the signed area of the parallelogram formed by the vectors (a1, a2) and (b1, b2).