Back To Course

Algebra II Textbook26 chapters | 256 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

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.

The **multiplicative inverse** of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. In math symbol speak, we have *A* * *A* sup -1 = *I*. This tells you that when you multiply a matrix *A* with its multiplicative inverse, you will get the identity matrix.

Yes, we write the inverse with a superscript of -1. When we deal with regular numbers, our multiplicative inverse is the number we multiply by to get 1. So, for the number 2, it is 1/2. For the number -3, the multiplicative inverse is 1/3. For our normal numbers, the multiplicative inverse is simply 1 divided by our number.

Unfortunately, not all matrices will have an inverse, nor is finding the multiplicative inverse that simple. In order to find the multiplicative inverse, we have to find the matrix for which, when we multiply it with our matrix, we get the identity matrix. Our matrices must also be square, having the same number of rows and columns.

We will leave the discussion on how to find the inverse of a matrix to another lesson. For this lesson, we will talk about its benefits. You see, it is useful to learn about the multiplicative inverse of a matrix because if we know it, then we can use it to help us solve equations with matrices in them.

For example, we can use it to solve a problem like this:

This matrix equation is in the form of *Ax* = *b*, where *A* is your coefficient matrix, *x* is your variable matrix, and *b* is your answer matrix. While we can use other methods to solve such a problem, if we know the multiplicative inverse of our coefficient matrix, then we can easily solve the problem by simply multiplying both sides by the inverse.

So, if we knew *A* sup -1, our answer would be *x* = *A* sup -1 * *b*. Yes, our answer would be our answer matrix, *b*, multiplied by the multiplicative inverse of our coefficient matrix. Let's look at how this works. For our coefficient matrix, we have this matrix as the multiplicative inverse matrix:

We can check whether this inverse is real or not by multiplying it with our coefficient matrix to see if we get the identity matrix. Multiplying the two matrices, we see that we do get the identity matrix:

We know for sure now that this inverse is the real inverse, and it works for us.

To use this inverse to help us find our answer, we simply multiply the inverse with the right side of our problem. So, we have this:

In math symbols, we have *x* = *A* sup -1 * *b*. We go ahead and multiply the matrices together. From the top row, we get 1(11) + -2(5) = 11 - 10 = 1. For the bottom row, we get -1/2(11) + 3/2(5) = -11/2 + 15/2 = 4 / 2 = 2. Now, we have this:

Our answer can then be easily found by just translating this back into equation form. We get *x* = 1 and *y* = 2. As you can see, using the inverse of a matrix to find our solution can be a very easy thing to do. I would use this method whenever you know what the inverse of a matrix is.

Let's review what we've learned. The **multiplicative inverse** of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. In math symbol speak, we have *A* * *A* sup -1 = *I*.

You can use the multiplicative inverse of a matrix to solve problems in the form of *Ax* = *b*, where *A* is your coefficient matrix, *x* is your variable matrix, and *b* is your answer, or constant, matrix.

If you know the inverse of a matrix, you can solve the problem by multiplying the inverse of the matrix with the answer matrix, *x* = *A* sup -1 * *b*. After you multiply, you can then easily find the answer by translating back to equation form.

Following this lesson, you should be able to:

- Define multiplicative inverse of a matrix
- Explain when and how you can use the multiplicative inverse of a matrix to solve problems

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
9 in chapter 10 of the course:

Back To Course

Algebra II Textbook26 chapters | 256 lessons

- What is a Matrix? 5:39
- How to Write an Augmented Matrix for a Linear System 4:21
- How to Perform Matrix Row Operations 5:08
- Matrix Notation, Equal Matrices & Math Operations with Matrices 6:52
- How to Solve Inverse Matrices 6:29
- How to Solve Linear Systems Using Gaussian Elimination 6:10
- How to Solve Linear Systems Using Gauss-Jordan Elimination 5:00
- Inconsistent and Dependent Systems: Using Gaussian Elimination 6:43
- Multiplicative Inverses of Matrices and Matrix Equations 4:31
- Solving Systems of Linear Equations in Two Variables Using Determinants 4:54
- Solving Systems of Linear Equations in Three Variables Using Determinants 7:41
- Using Cramer's Rule with Inconsistent and Dependent Systems 4:05
- How to Evaluate Higher-Order Determinants in Algebra 7:59
- Go to Algebra II: Matrices and Determinants

- AFOQT Information Guide
- ACT Information Guide
- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- FTCE Middle Grades Math: Connecting Math Concepts
- Social Justice Goals in Social Work
- Developmental Abnormalities
- Overview of Human Growth & Development
- ACT Informational Resources
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- What Are the 5 Ws in Writing? - Uses & Examples
- Phenol: Preparation & Reactions
- What is a Color Wheel? - Definition & Types
- What Are Abbreviations? - Meaning, Types & Examples
- Zentangle Lesson Plan for High School
- West Side Story Discussion Questions
- Fireboat: The Heroic Adventures of the John J. Harvey Activities
- Quiz & Worksheet - Solvay Process
- Quiz & Worksheet - Acetone Reactions
- Quiz & Worksheet - Themes in A Raisin in the Sun
- Quiz & Worksheet - Act & Rule Utilitarianism Comparison
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Rubrics | Rubric Examples and Templates
- Calculus Worksheets

- How to Prepare for a Job Interview
- High School Psychology: Homeschool Curriculum
- How to Lead a Customer Service Team
- CLEP Introductory Business Law: Study Guide & Test Prep
- AP Environmental Science: Homeschool Curriculum
- TExMaT Master Mathematics Teacher 8-12: Integration Techniques
- TExMaT Master Mathematics Teacher 8-12: Algebraic Expressions
- Quiz & Worksheet - Self-Identity in Children
- Quiz & Worksheet - Promoting Literacy in the Classroom
- Quiz & Worksheet - Freud's Theory of Psychoanalysis
- Quiz & Worksheet - Motivation Theories
- Quiz & Worksheet - Characteristics of Collaborative Skills

- Business Case Study: Communication at Dell
- Brainstorming Lesson Plan for Elementary School
- Behavior Rubric Examples
- Rules for the QTS Numeracy Skills Test Candidates
- Florida State Standards for Math
- Common Core State Standards in Colorado
- What Are the NGSS Cross Cutting Concepts?
- What Is The Difference Between NGSS & CCSS?
- eBooks vs. Textbooks
- Parent, Student & Teacher Contract
- What are the NBPTS Core Propositions?
- States that Require Physical Education

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject