# Ch 18: Algebra: Determinants & Transformations

### About This Chapter

## Algebra: Determinants & Transformations - Chapter Summary

Tackle the lessons in this chapter to help you understand determinants and transformations in linear algebra. Instructors use engaging narration to explain the material and provide examples and solutions to illustrate how these equations, matrices and theorems work. The lessons cover the following topics:

- Use of the Laplace expansion equation to compute determinants
- Formation of the adjugate matrix and the standard matrix
- Explanation of how to use Cramer's Rule to find solutions of a square linear system
- Examples of eigenvalues and eigenvectors
- The Cayley-Hamilton Theorem and the Rank-Nullity Theorem

Taught by math experts, these brief video lessons break difficult subject matter into easy-to-grasp components to ensure you understand the material. If you need to review the lessons, you can either watch the entire video again or use video tags to jump to key points within the video. Transcripts provide an alternative way to bolster what you have learned. Be sure to take the lesson quizzes and chapter exam as a means of self-assessment.

### 1. Laplace Expansion Equation & Finding Determinants

The Laplace Expansion equation (LEE) applies determinants of smaller matrices to a larger square matrix to identify the determinant. Analyze the LLE method to break down the equation into mathematical operations and apply it to the so-called checkerboard where N = 1, 2, or 3.

### 2. Adjugate Matrix: Definition, Formation & Example

An adjugate matrix identifies applications and is the transpose of a cofactor matrix found by dividing the adjugate matrix by the determinant. Learn how adjugate matrixes are formed mathematically using minors, determinants, and transposes.

### 3. Using Cramer's Rule with Inconsistent and Dependent Systems

Cramer's rule is a very acute way to solve linear systems for its various variables. Learn more about using Cramer's rule with inconsistent and dependent systems.

### 4. Eigenvalues & Eigenvectors: Definition, Equation & Examples

In mathematics, eigenvalues and eigenvectors are special values found in a square matrix. Explore the definition, equation, and examples of eigenvalues and eigenvectors. Understand how to find them, and recognize representative eigenvectors.

### 5. Cayley-Hamilton Theorem Definition, Equation & Example

The Cayley-Hamilton theorem shows how a matrix's special polynomial always equals 0. Explore this definition, the theorem history, and how it works, with examples of how to solve for a matrix inverse using the theorem equation.

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### Other Chapters

Other chapters within the GRE Math: Study Guide & Test Prep course

- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Definite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Analytic Geometry in 3-Dimensions
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
- GRE Math Flashcards