Chapter 3: Determinants and Diagonalization

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3.1 The Cofactor Expansion

Example 3.1.1

Definition 3.1 Cofactors of a Matrix

Example 3.1.2

Definition 3.2 Cofactor expansion of a Matrix

Theorem 3.1.1: Cofactor Expansion Theorem2

Example 3.1.3

Example 3.1.4

Theorem 3.1.2

Example 3.1.5

Example 3.1.6

Example 3.1.7

Example 3.1.8

Theorem 3.1.3

Example 3.1.9

Theorem 3.1.4

Theorem 3.1.5

Example 3.1.10

Theorem 3.1.6

Exercises for 3.1

Exercise 3.1.8 Show that:

Exercise 3.1.13

Exercise 3.1.15

3.2 Determinants and Matrix Inverses

Theorem 3.2.1: Product Theorem

Example 3.2.1

Theorem 3.2.2

Example 3.2.2

Example 3.2.3

Theorem 3.2.3

Example 3.2.4

Example 3.2.5

Adjugates

Definition 3.3 Adjugate of a Matrix

Example 3.2.6

Theorem 3.2.4: Adjugate Formula

Example 3.2.7

Example 3.2.8

Cramer’s Rule

Theorem 3.2.5: Cramer’s Rule5

Example 3.2.9

Polynomial Interpolation

Example 3.2.10

Solution.

Theorem 3.2.6

Theorem 3.2.7

Exercises for 3.2

Exercise 3.2.26

3.3 Diagonalization and Eigenvalues

Example 3.3.1

Theorem 3.3.1

Eigenvalues and Eigenvectors

Definition 3.4 Eigenvalues and Eigenvectors of a Matrix

Example 3.3.2

Definition 3.5 Characteristic Polynomial of a Matrix

Theorem 3.3.2

Example 3.3.3

Example 3.3.4

Example 3.3.5

A-Invariance

Example 3.3.6

Theorem 3.3.3

Example 3.3.7

Solution.

Diagonalization

Definition 3.6 Diagonalizable Matrices

Theorem 3.3.4

Example 3.3.8

Example 3.3.9

Definition 3.7 Multiplicity of an Eigenvalue

Theorem 3.3.5

Theorem 3.3.6

Diagonalization Algorithm

Example 3.3.10

Example 3.3.11

Linear Dynamical Systems

Example 3.3.12

Definition 3.8 Linear Dynamical System

Theorem 3.3.7

Example 3.3.13

Example 3.3.14

Graphical Description of Dynamical Systems

Example 3.3.15

Example 3.3.16

Example 3.3.17

Example 3.3.18

Google PageRank

Exercises for 3.3

Exercise 3.3.9

Exercise 3.3.19

Exercise 3.3.27

3.4 An Application to Linear Recurrences

Example 3.4.1

Example 3.4.2

Example 3.4.3

Example 3.4.4

Exercises for 3.4

3.5 An Application to Systems of Differential Equations

The Exponential Function

Theorem 3.5.1

Example 3.5.1

General Differential Systems

Example 3.5.2

Theorem 3.5.2

Example 3.5.3

Exercises for 3.5

3.6 Proof of the Cofactor Expansion Theorem

Lemma 3.6.1

Lemma 3.6.2

Theorem 3.6.1

Exercises for 3.6

Supplementary Exercises for Chapter 3

Section worksheets