# Ch 10: Algebra II: Matrices and Determinants

### About This Chapter

## Algebra II: Matrices and Determinants - Chapter Summary and Learning Objectives

If you're having a hard time wrapping your head around matrices, this chapter is for you. Our instructors can show you how these sets of numbers are used to solve systems of equations in linear algebra, making matrices a useful tool in a variety of fields, including computer programming. Watch the videos to learn how to perform basic arithmetic operations with matrices and work with their determinants. By the end of this chapter, you should be able to:

- Write matrices and perform matrix row operations
- Add, subtract and multiply matrices and find their inverses
- Apply Gaussian and Gauss-Jordan elimination methods
- Find a matrix's determinant and use it to solve a linear system

Video | Objectives |
---|---|

What Is a Matrix? | Identify the components of a matrix and explore its uses to solve systems of equations. Find out how to add, subtract and multiply matrices. |

How to Write an Augmented Matrix for a Linear System | Learn the steps involved in writing an augmented matrix for a linear system. |

How to Perform Matrix Row Operations | Understand how to perform basic matrix row operations. |

Matrix Notation, Equal Matrices & Math Operations with Matrices | Learn what equal matrices are and demonstrate the proper use of matrix notation. |

How to Solve Inverse Matrices | Find out what it takes to identify and solve inverse matrices. |

How to Solve Linear Systems Using Gaussian Elimination | Understand how to solve linear systems using matrices and Gaussian elimination. |

How to Solve Linear Systems Using Gauss-Jordan Elimination | Explain how to use matrices and Gauss-Jordan elimination to solve linear systems. |

Inconsistent and Dependent Systems: Using Gaussian Elimination | Use Gaussian elimination to solve linear systems with more variables than equations or linear systems without unique solutions. |

Multiplicative Inverses of Matrices and Matrix Equations | Find the multiplicative inverse of a square matrix. Learn how inverses can be used to solve matrix equations and encode and decode messages. |

How to Take a Determinant of a Matrix | Identify square matrices and discover methods for finding their determinants. |

Solving Systems of Linear Equations in Two Variables Using Determinants | Find out how to evaluate a second-order determinant. Use Cramer's rule to solve a system of linear equations in two variables. |

Solving Systems of Linear Equations in Three Variables Using Determinants | Evaluate a third-order determinant. Understand the steps required to solve a system of linear equations in three variables using Cramer's rule. |

Using Cramer's Rule with Inconsistent and Dependent Systems | Identify linear systems with dependent equations as well as inconsistent systems using determinants. |

How to Evaluate Higher-Order Determinants in Algebra | Explain the processes involved in evaluating higher-order determinants. |

### 1. What is a Matrix?

As math gets more and more complicated and there become more and more numbers flying around, it becomes really handy to put all these numbers in a nice organized grid... hello matrices! Learn about what they are and why there are used.

### 2. How to Write an Augmented Matrix for a Linear System

Watch this video lesson to learn how you can turn your system of equations into matrix form. The matrix form is another way of writing your linear system that is sometimes easier to work with.

### 3. How to Perform Matrix Row Operations

Watch this video lesson to learn how easy it is to perform row operations on a matrix. Learn how to perform the three basic operations easily and quickly.

### 4. Matrix Notation, Equal Matrices & Math Operations with Matrices

Watch this video lesson to learn about the basics of matrices and what kinds of math operations you can do with them. Also learn how to determine whether two matrices are the same or not.

### 5. How to Solve Inverse Matrices

Watch this video lesson to learn what kinds of matrix operations you can take to find the inverse of a matrix. Also learn why matrix inverses are useful.

### 6. How to Solve Linear Systems Using Gaussian Elimination

Watch this video lesson to learn an easy way to solve a system of equations that involves manipulating a matrix. Learn the kinds of easy matrix manipulations that are needed to solve any system of equations.

### 7. How to Solve Linear Systems Using Gauss-Jordan Elimination

You will come across simple linear systems and more complex ones as you progress in math. Watch this video lesson to learn how you can use Gauss-Jordan elimination to help you solve these linear systems.

### 8. Inconsistent and Dependent Systems: Using Gaussian Elimination

Watch this video lesson to learn whether or not you can use Gaussian elimination to solve inconsistent and dependent systems. Also, learn whether there is another way to find solutions of these systems.

### 9. Multiplicative Inverses of Matrices and Matrix Equations

Watch this video lesson to learn about another method you can use to solve a matrix problem if you are given the inverse of the matrix. You will also learn the identifying mark of the multiplicative inverse of a matrix.

### 10. How to Take a Determinant of a Matrix

Matrices are incredibly powerful and can help you do all sorts of things, but one of the most basic (and surprisingly helpful) operations you can perform on one is to take its determinant. Learn how to do that here!

### 11. Solving Systems of Linear Equations in Two Variables Using Determinants

Watch this video lesson to learn how you can easily find the solution of a linear system in two variables by just using determinants. All you need to find the complete solution is to find three determinants.

### 12. Solving Systems of Linear Equations in Three Variables Using Determinants

Watch this video lesson to learn how to use Cramer's rule to help you find your solutions by finding just four determinants. Learn the numbers to use for your determinants.

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

Watch this video lesson, and you will see what happens when we use Cramer's Rule with inconsistent and dependent systems. You will see what kind of result you will always get when you try to use Cramer's Rule.

### 14. How to Evaluate Higher-Order Determinants in Algebra

Watch this video lesson to learn the rules for taking the determinant of any size matrix. Learn the kind of matrix that you need before you can take the determinant. Watch as the pattern of the determinant takes shape.

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### Other Chapters

Other chapters within the Algebra II Textbook course

- Algebra II: Basic Arithmetic Review
- Algebra II: Algebraic Expressions and Equations Review
- Algebra II: Real Numbers
- Algebra II: Complex and Imaginary Numbers Review
- Algebra II: Exponents and Exponential Expressions Review
- Algebra II: Properties of Functions Review
- Algebra II: Linear Equations Review
- Algebra II: Systems of Linear Equations
- Algebra II: Inequalities Review
- Algebra II: Absolute Value Review
- Algebra II: Polynomials
- Algebra II: Factoring
- Algebra II: Graphing and Factoring Quadratic Equations Review
- Algebra II: Rational Expressions
- Algebra II: Graphing and Functions
- Algebra II: Roots and Radical Expressions Review
- Algebra II: Quadratic Equations
- Algebra II: Exponential and Logarithmic Functions
- Algebra II: Conic Sections
- Algebra II: Sequences and Series
- Algebra II: Sets
- Algebra II: Combinatorics
- Algebra II: Calculations, Ratios, Percent & Proportions Review
- Algebra II: Statistics
- Algebra II: Trigonometry