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4: Determinants

Last updated

Mar 16, 2025
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- 3.6: The Invertible Matrix Theorem
- 4.1: Determinants- Definition

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Objectives

Learn about determinants, their computation, and their properties.

We begin by recalling the overall structure of this book:

Solve the matrix equation $Ax=b$ .
Solve the matrix equation $Ax=\lambda x\text{,}$ where $\lambda$ is a number.
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.

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
This page provides an extensive overview of determinants in linear algebra, detailing their definitions, properties, and computation methods, particularly through row reduction. It emphasizes the significance of row operations, such as swaps and scaling, and introduces concepts like triangular matrices and multilinearity. Key properties include conditions for invertibility, the relationship between determinants of products, transposes, and the implications of zero determinants.
4.2: Cofactor Expansions
This page explores various methods for computing the determinant of matrices, primarily using cofactor expansions. It covers the properties of determinants, like multilinearity and invariance under transposition, and describes the relationship between cofactors and inverses. The text illustrates the recursive nature of determinants through examples, including the efficiency of cofactor expansion with zeros.
4.3: Determinants and Volumes
This page explores the connections between matrices, their determinants, and geometric volumes, focusing on parallelepipeds. It defines parallelepipeds and illustrates how the determinant relates to their volume, covering key determinant properties and their impact on area and volume calculations for parallelograms and triangles. The text demonstrates how linear transformations scale volumes by the absolute value of the determinant.

Objectives

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