**Matrix Algebra**

## Matrix Operations and Determinants

1 hr 10 min 14 Examples

- Basic Algebraic Matrix Operations
- Properties and Definitions of Matrix Operations
- Examples (#1-4) for performing matrix operations
- Examples (#5-8) for performing matrix operations
- Example #9 multiplying two matrices
- Example #10 multiplying two matrices
- Powers of a Matrix with Example
- Overview of how to calculate a Determinant of a Matrix
- Two Examples of how to calculate a 2×2 Determinant
- Example #1 of how to calculate a 3×3 Determinant
- Example #2 of how to calculate a 3×3 Determinant
- Theorem and Properties for the Sum and Products of a Transpose Matrix

## Inverse Matrix

59 min 11 Examples

- Overview and Definition of an Invertible Matrix
- Theorem and Formula for Calculating an Inverse with Three Examples
- Overview of the Algorithm for Finding Inverse Matrices
- Example #1 Find the Inverse of a 2×2 matrix using the Algorithm
- Example #2 Find the Inverse of a 3×3 matrix using the Algorithm
- Overview and Proof of How to Solve a System of Linear Equations using Inverses
- Example #1 use the inverse to solve for a specific matrix
- Example #2 use the inverse to solve a system
- Example #3 use the inverse to solve a system
- Facts and Definition of Elementary Matrices
- Three Examples of Finding the Inverses of Elementary Matrices

## Invertible Matrix Theorem

34 min 10 Examples

- Facts about the Inverse Matrices with Examples
- Example – find matrix A using inverses
- The Invertible Matrix Theorem
- Five True/False Questions for using the Invertible Matrix Theorem
- Three Examples for using the Invertible Matrix Theorem given certain implications
- Four Examples using the Invertible Matrix Theorem to quickly determine if a Matrix is Invertible
- Understanding Invertible Linear Transformations for an Invertible Matrix

## Partitioned Matrices

51 min 8 Examples

- What is a Partitioned Matrix?
- Two Examples finding the subdivisions or blocks and dimensions of a Partitioned Matrix
- Overview of how to Add and Multiply a Partitioned Matrix
- Examples of Multiplying two Partitioned Matrices
- Theorem of Column-Row Expansion of AB
- Three Examples of Comfortably Partitioned Block Multiplication
- Example #1 – Solving Partitoned Matrix Equation
- Example #2 – Solving Partitoned Matrix Equation
- Overview of Block Diagonal Matrices and Inverses of Partitioned Matrices

## Matrix Factorization

1 hr 7 min 5 Examples

- LU-Factorization and its Applications
- Steps for LU-Factorization
- Example #1 – find the LU-Factorization using Method 1
- Example #2 – find the LU-Factorization using Method 2
- Example #3 – find the LU Factorization using Method 1 and Method 2
- Steps for Solve a System of Linear Equations using LU-Decomposition
- Example #1 – solve the matrix equation using LU Factorization
- Example #2 – solve the matrix equation using LU Factorization

## Applications to Computer Graphics

52 min 6 Examples

- Overview of the Applications to Computer Graphics
- Review of Basic Transformations and Intro to Homogeneous Coordinates
- Composite Transformation Algorithm
- Example #1 find new coordinates for the given composite 2D transformation
- Example #2 find new coordinates for the given composite 2D transformation
- Example #3 find a 3×3 matrix using composite 2D transformations and Homogeneous Coordinates
- Example #4 find a 3×3 matrix using composite 2D transformations and Homogeneous Coordinates
- Example #5 find a 3×3 matrix using composite 2D transformations and Homogeneous Coordinates
- Homogeneous 3D Coordinates and Perspective Projections Matrix
- Example – find the image under a given Perspective Projection