##### Primary Goal

Learn about determinants: their computation and their properties.

We begin by recalling the overall structure of this book:

- Solve the matrix equation
- Solve the matrix equation where is a number.
- Approximately solve the matrix equation

At this point we have said all that we will say about the first part. This chapter belongs to the second.

Learn about determinants: their computation and their properties.

The *determinant* of a square matrix is a number This incredible quantity is one of the most important invariants of a matrix; as such, it forms the basis of most advanced computations involving matrices.

In Section 5.1, we will define the determinant in terms of its behavior with respect to row operations. The determinant satisfies many wonderful properties: for instance, if and only if is invertible. We will discuss some of these properties in Section 5.1 as well. In Section 5.2, we will give a recursive formula for the determinant of a matrix. This formula is very useful, for instance, when taking the determinant of a matrix with unknown entries; this will be important in Chapter 6. Finally, in Section 5.3, we will relate determinants to volumes. This gives a geometric interpretation for determinants, and explains why the determinant is defined the way it is. This interpretation of determinants is a crucial ingredient in the change-of-variables formula in multivariable calculus.