In this chapter we deal with the determinant of a square matrix. The determinant has a simple geometric meaning, that of signed volume, and we use that to develop it in Section 3.1. We then present a more traditional and fuller development in Section 3.2. In Section 3.3 we derive important and useful properties of the determinant. In Section 3.4 we consider integrality questions, e.g., the question of the existence of integer (not just rational) solutions of the linear system Ax = b, a question best answered using determinants. In Section 3.5 we consider orientations, and see how to explain the meaning of the sign of the determinant in the case of real vector spaces. In Section 3.6 we present an interesting family of examples, the Hilbert matrices.

THE GEOMETRY OF VOLUMES

The determinant of a matrix A has a simple geometric meaning. It is the (signed) volume of the image of the unit cube under the linear transformation JA.

We will begin by doing some elementary geometry to see what properties (signed) volume should have, and use that as the basis for the not-so simple algebraic definition.

Henceforth we drop the word “signed” and just refer to volume.

In considering properties that volume should have, suppose we are working in ℝ2, where volume is area. Let A be the matrix A = [v1 | v2]. The unit square in ℝ2 is the parallelogram determined by the standard unit vectors e1 and e2.