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 Subscript notation for the matrix A The row 1, column 1 element is a 11 ; the row 2, column 3 element is a 23 ; and, in general, the row i, column j element is a ij. 7.5 Determinants and Cramer’s Rule

4. Copyright © 2007 Pearson Education, Inc. Slide 7-4 7.5 Determinants of 2 × 2 Matrices Associated with every square matrix is a real number called the determinant of A. In this text, we use det A. The determinant of a 2 × 2 matrix A, is defined as

5. Copyright © 2007 Pearson Education, Inc. Slide 7-5 ExampleFind det A if Analytic Solution Graphing Calculator Solution 7.5 Determinants of 2 × 2 Matrices

6. Copyright © 2007 Pearson Education, Inc. Slide 7-6 7.5 Determinant of a 3 × 3 Matrix The determinant of a 3 × 3 matrix A, is defined as

7. Copyright © 2007 Pearson Education, Inc. Slide 7-7 A method for calculating 3 × 3 determinants is found by re-arranging and factoring this formula. Each of the quantities in parentheses represents the determinant of a 2 × 2 matrix that is part of the 3 × 3 matrix remaining when the row and column of the multiplier are eliminated. 7.5 Determinant of a 3 × 3 Matrix

8. Copyright © 2007 Pearson Education, Inc. Slide 7-8 7.5 The Minor of an Element The determinant of each 3 × 3 matrix is called a minor of the associated element. The symbol M ij represents the minor when the ith row and jth column are eliminated.

9. Copyright © 2007 Pearson Education, Inc. Slide 7-9 To find the determinant of a 3 × 3 or larger square matrix: 1.Choose any row or column, 2.Multiply the minor of each element in that row or column by a +1 or –1, depending on whether the sum of i + j is even or odd, 3.Then, multiply each cofactor by its corresponding element in the matrix and find the sum of these products. This sum is the determinant of the matrix. 7.5 The Cofactor of an Element Let M ij be the minor for element a ij in an n × n matrix. The cofactor of a ij, written A ij, is

10. Copyright © 2007 Pearson Education, Inc. Slide 7-10 ExampleEvaluate det, expanding by the second column. SolutionFirst find the minors of each element in the second column. 7.5 Finding the Determinant

11. Copyright © 2007 Pearson Education, Inc. Slide 7-11 Now, find the cofactor. The determinant is found by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products. 7.5 Finding the Determinant

12. Copyright © 2007 Pearson Education, Inc. Slide 7-12 7.5 Cramer’s Rule for 2 × 2 Systems Note: Cramer’s rule does not apply if D = 0. For the system where, if possible,

13. Copyright © 2007 Pearson Education, Inc. Slide 7-13 ExampleUse Cramer’s rule to solve the system. Analytic Solution By Cramer’s rule, 7.5 Applying Cramer’s Rule to a System with Two Equations

14. Copyright © 2007 Pearson Education, Inc. Slide 7-14 The solution set is Graphing Calculator Solution Enter D, D x, and D y as matrices A, B, and C, respectively. 7.5 Applying Cramer’s Rule to a System with Two Equations

15. Copyright © 2007 Pearson Education, Inc. Slide 7-15 7.5 Cramer’s Rule for a System with Three Equations For the system where