1. Copyright © 2007 Pearson Education, Inc. Slide 7-1

2. Copyright © 2007 Pearson Education, Inc. Slide 7-2 Chapter 7: Matrices and Systems of Equations and Inequalities 7.1Systems of Equations 7.2Solution of Linear Systems in Three Variables 7.3Solution of Linear Systems by Row Transformations 7.4Matrix Properties and Operations 7.5Determinants and Cramer’s Rule 7.6Solution of Linear Systems by Matrix Inverses 7.7Systems of Inequalities and Linear Programming 7.8Partial Fractions

3. Copyright © 2007 Pearson Education, Inc. Slide 7-3 7.6 Solution of Linear Systems by Matrix Inverses If there is a multiplicative identity matrix I, such that for any matrix A, then A and I must be square matrices of the same dimensions. The 2 × 2 identity matrix, denoted I 2, is This is easily verified by showing A I 2 = A and I 2 A = A, for any 2 × 2 matrix A.

4. Copyright © 2007 Pearson Education, Inc. Slide 7-4 ExampleShow that Graphing Calculator Solution Using 7.6 Using the 3 × 3 Identity Matrix I 3

5. Copyright © 2007 Pearson Education, Inc. Slide 7-5 Suppose If AB = I 2 and BA = I 2, then B is the inverse of A. Inverse of a 2 × 2 matrix 7.6 Multiplicative Inverses of Square Matrices

6. Copyright © 2007 Pearson Education, Inc. Slide 7-6 Solving the system yields We can show that AB = I 2 and BA = I 2.. Thus, we can conclude that B is the inverse of A, written A -1, provided that the det A  0. 7.6 The Inverse of a 2 × 2 Matrix

7. Copyright © 2007 Pearson Education, Inc. Slide 7-7 7.6 The Inverse of a 2 × 2 Matrix If and det A  0, then or

8. Copyright © 2007 Pearson Education, Inc. Slide 7-8 ExampleFind A -1 if it exists. Analytic Solution (a) (b) Here, A -1 does not exist. 7.6 Finding the Inverse of a 2 × 2 Matrix

9. Copyright © 2007 Pearson Education, Inc. Slide 7-9 Graphing Calculator Solution (a) (b) 7.6 Finding the Inverse of a 2 × 2 Matrix The calculator returns a singular matrix error when directed to find the inverse of a matrix whose determinant is 0.

10. Copyright © 2007 Pearson Education, Inc. Slide 7-10 Solve the system AX = B, where A is the coefficient matrix, X is the matrix of variables, and B is the matrix of the constants. Note: When multiplying by matrices on both sides, multiply in the same order on both sides. 7.6 Solving Linear Systems Using Inverse Matrices Multiply both sides by A -1. Associative property Multiplicative inverse property Identity property

11. Copyright © 2007 Pearson Education, Inc. Slide 7-11 7.6 Solving Linear Systems Using Inverse Matrices ExampleSolve the system using the inverse of the coefficient matrix. Analytic Solution

12. Copyright © 2007 Pearson Education, Inc. Slide 7-12 The solution set {(x, y, z)} = {(4, 2, –3)}. 7.6 Solving Linear Systems Using Inverse Matrices

13. Copyright © 2007 Pearson Education, Inc. Slide 7-13 Graphing Calculator Solution Enter the coefficient matrix A and the constant matrix B. Make sure the det A  0. The solution verifies the results achieved analytically. 7.6 Solving Linear Systems Using Inverse Matrices