
# 4: Determinants


We begin by recalling the overall structure of this book:

1. Solve the matrix equation $$Ax=b$$.
2. Solve the matrix equation $$Ax=\lambda x\text{,}$$ where $$\lambda$$ 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.

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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 4.1, we will define the determinant in terms of its behavior with respect to row operations. The determinant satisfies many wonderful properties: for instance, $$\det(A) \neq 0$$ if and only if $$A$$ is invertible. We will discuss some of these properties in Section 4.1 as well. In Section 4.2, 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 5. Finally, in Section 4.3, 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.

• 4.1: Determinants- Definition
In this section, we define the determinant, and we present one way to compute it. Then we discuss some of the many wonderful properties the determinant enjoys.
• 4.2: Cofactor Expansions
In this section, we give a recursive formula for the determinant of a matrix, called a cofactor expansion. The formula is recursive in that we will compute the determinant of an n×n matrix assuming we already know how to compute the determinant of an (n−1)×(n−1) matrix. At the end is a supplementary subsection on Cramer’s rule and a cofactor formula for the inverse of a matrix.
• 4.3: Determinants and Volumes
In this section we give a geometric interpretation of determinants, in terms of volumes. This will shed light on the reason behind three of the four defining properties of the determinant. It is also a crucial ingredient in the change-of-variables formula in multivariable calculus.