Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations.

Slides:



Advertisements
Similar presentations
1.5 Elementary Matrices and a Method for Finding
Advertisements

Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations.
Chapter 4 Systems of Linear Equations; Matrices Section 6 Matrix Equations and Systems of Linear Equations.
Matrices: Inverse Matrix
1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-1 Systems of Equations and Inequalities Chapter 4.
Matrix Algebra THE INVERSE OF A MATRIX © 2012 Pearson Education, Inc.
Chapter 2 Section 2. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. The Multiplication Property of Equality Use the multiplication.
EXAMPLE 2 Solve a matrix equation SOLUTION Begin by finding the inverse of A = Solve the matrix equation AX = B for the 2 × 2 matrix X. 2 –7 –1.
Copyright © Cengage Learning. All rights reserved. 7.6 The Inverse of a Square Matrix.
Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations.
Copyright © 2007 Pearson Education, Inc. Slide 7-2 Chapter 7: Systems of Equations and Inequalities; Matrices 7.1Systems of Equations 7.2Solution of Linear.
1 1.1 © 2012 Pearson Education, Inc. Linear Equations in Linear Algebra SYSTEMS OF LINEAR EQUATIONS.
Systems of Equations and Inequalities
3.5 Solution by Determinants. The Determinant of a Matrix The determinant of a matrix A is denoted by |A|. Determinants exist only for square matrices.
Slide 5-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Copyright © 2007 Pearson Education, Inc. Slide 7-1.
Rev.S08 MAC 1140 Module 10 System of Equations and Inequalities II.
Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Eight Systems: Matrices.
Copyright © 2011 Pearson, Inc. 7.3 Multivariate Linear Systems and Row Operations.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 9 Matrices and Determinants.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
4.6 Matrix Equations and Systems of Linear Equations In this section, you will study matrix equations and how to use them to solve systems of linear equations.
Copyright © 2007 Pearson Education, Inc. Slide 7-1.
Copyright © 2011 Pearson Education, Inc. Slide
Copyright © 2007 Pearson Education, Inc. Slide 7-1.
Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations.
Chapter 9 Matrices and Determinants Copyright © 2014, 2010, 2007 Pearson Education, Inc Multiplicative Inverses of Matrices and Matrix Equations.
8.1 Matrices & Systems of Equations
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 5 Systems and Matrices Copyright © 2013, 2009, 2005 Pearson Education, Inc.
1 C ollege A lgebra Systems and Matrices (Chapter5) 1.
Warm-Up 3) Find the determinant by hand. 4) Find the determinant using your calculator. 1) Multiply. Show work. 2) Multiply. Show work.
Inverse and Identity Matrices Can only be used for square matrices. (2x2, 3x3, etc.)
Have we ever seen this phenomenon before? Let’s do some quick multiplication…
CW Matrix Division We have seen that for 2x2 (“two by two”) matrices A and B then AB  BA To divide matrices we need to define what we mean by division!
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 1.
Copyright © 2011 Pearson, Inc. P.3 Linear Equations and Inequalities.
Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations.
Copyright © Cengage Learning. All rights reserved. 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES Read pp Stop at “Inverse of a Matrix” box.
Chapter 2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-1 Solving Linear.
Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 8-1 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 8 Systems of Linear Equations.
Copyright © 2007 Pearson Education, Inc. Slide 7-1.
1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-1 Systems of Equations and Inequalities Chapter 4.
TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Chapter 5 More Work with Matrices
Chapter 7 Solving systems of equations substitution (7-1) elimination (7-1) graphically (7-1) augmented matrix (7-3) inverse matrix (7-3) Cramer’s Rule.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.2 – Slide 1.
Copyright © 2011 Pearson Education, Inc. Slide
2 2.2 © 2016 Pearson Education, Ltd. Matrix Algebra THE INVERSE OF A MATRIX.
Slide Copyright © 2009 Pearson Education, Inc. 7.4 Solving Systems of Equations by Using Matrices.
Copyright © Cengage Learning. All rights reserved. 8 Matrices and Determinants.
College Algebra Chapter 6 Matrices and Determinants and Applications
Use Inverse Matrices to Solve Linear Systems
TYPES OF SOLUTIONS SOLVING EQUATIONS
Chapter 4 Systems of Linear Equations; Matrices
TYPES OF SOLUTIONS SOLVING EQUATIONS
The Inverse of a Square Matrix
10.5 Inverses of Matrices and Matrix Equations
Inverse of a Square Matrix
Use Inverse Matrices to Solve Linear Systems
Chapter 7: Matrices and Systems of Equations and Inequalities
Multiplicative Inverses of Matrices and Matrix Equations
Use Inverse Matrices to Solve 2 Variable Linear Systems
Section 9.5 Inverses of Matrices
Bellwork 1) Multiply. 3) Find the determinant. 2) Multiply.
Sec 3.5 Inverses of Matrices
1.11 Use Inverse Matrices to Solve Linear Systems
Matrix Algebra THE INVERSE OF A MATRIX © 2012 Pearson Education, Inc.
Solving Linear Systems of Equations - Inverse Matrix
Presentation transcript:

Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables 9.3 Matrices and Systems of Equations 9.4 Matrix Operations 9.5 Inverses of Matrices 9.6 Determinants and Cramer’s Rule 9.7 Systems of Inequalities and Linear Programming 9.8 Partial Fractions

Copyright © 2009 Pearson Education, Inc. 9.5 Inverses of Matrices  Find the inverse of a square matrix, if it exists.  Use inverses of matrices to solve systems of equations.

Slide Copyright © 2009 Pearson Education, Inc. The Identity Matrix

Slide Copyright © 2009 Pearson Education, Inc. Example For find each of the following. a) AI b) IA

Slide Copyright © 2009 Pearson Education, Inc. Inverse of a Matrix For an n  n matrix A, if there is a matrix A  1 for which A  1 A = I = A A  1, then A  1 is the inverse of A. Verify that is the inverse of. We show that BA = I = AB.

Slide Copyright © 2009 Pearson Education, Inc. Finding an Inverse Matrix To find an inverse, we first form an augmented matrix consisting of A on the left side and the identity matrix on the right side. Then we attempt to transform the augmented matrix to one of the form The 2  2 identity matrix The 2  2 matrix A

Slide Copyright © 2009 Pearson Education, Inc. Example Find A  1, where A =.

Slide Copyright © 2009 Pearson Education, Inc. Example continued Thus, A  1 =.

Slide Copyright © 2009 Pearson Education, Inc. Notes If a matrix has an inverse, we say that it is invertible, or nonsingular. When we cannot obtain the identity matrix on the left using the Gauss-Jordan method, then no inverse exists.

Slide Copyright © 2009 Pearson Education, Inc. Solving Systems of Equations Matrix Solutions of Systems of Equations For a system of n linear equations in n variables, AX = B, if A is an invertible matrix, then the unique solution of the system is given by X = A  1 B. Since matrix multiplication is not commutative in general, care must be taken to multiply on the left by A  1.

Slide Copyright © 2009 Pearson Education, Inc. Example Use an inverse matrix to solve the following system of equations: 3x + 4y = 5 5x + 7y = 9 We write an equivalent matrix, AX = B: In the previous example we found A  1 =

Slide Copyright © 2009 Pearson Education, Inc. Example continued We now have X = A  1 B. The solution of the system of equations is (  1, 2).