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We begin by recalling the overall structure of this book:

  1. Solve the matrix equation Ax = b .
  2. Solve the matrix equation Ax = λ x , where λ is a number.
  3. Approximately solve the matrix equation Ax = b .

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

Primary Goal

Learn about determinants: their computation and their properties.

The determinant of a square matrix A is a number det ( A ) . 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, det ( A ) A = 0 if and only if A 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.