1. Chapter 9 Matrices and Determinants Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 9.5 Determinants and Cramer’s Rule

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Evaluate a second-order determinant. Solve a system of linear equations in two variables using Cramer’s Rule. Evaluate a third-order determinant. Solve a system of linear equations in three variables using Cramer’s Rule. Evaluate higher-order determinants. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Determinant of a 2 x 2 Matrix

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Evaluating the Determinant of a 2 x 2 Matrix Evaluate the determinant of the following matrix:

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Evaluating the Determinant of a 2 x 2 Matrix Evaluate the determinant of the following matrix:

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Solving a Linear System in Two Variables Using Determinants

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Using Cramer’s Rule to Solve a Linear System Use Cramer’s Rule to solve the system: 1. 2.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Using Cramer’s Rule to Solve a Linear System (continued) Use Cramer’s Rule to solve the system: 3. 4. The solution set is

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 The Determinant of a 3 x 3 Matrix

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Definition of the Determinant of a 3 x 3 Matrix

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Evaluating the Determinant of a 3 x 3 Matrix

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Evaluating the Determinant of a 3 x 3 Matrix Evaluate the determinant of the following matrix: We find the minor for each numerical factor: The numerical factors are highlighted. The minor for 2 is

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Evaluating the Determinant of a 3 x 3 Matrix (continued) Evaluate the determinant of the following matrix: We find the minor for each numerical factor: The numerical factors are highlighted. The minor for – 5 is

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Evaluating the Determinant of a 3 x 3 Matrix (continued) Evaluate the determinant of the following matrix: We find the minor for each numerical factor: The numerical factors are highlighted. The minor for –4 is

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Evaluating the Determinant of a 3 x 3 Matrix (continued) Evaluate the determinant of the following matrix: We multiply each numerical factor by its second-order determinant and calculate.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Solving Systems of Linear Equations in Three Variables Using Determinants

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Solving Three Equations in Three Variables Using Determinants – Cramer’s Rule (continued)

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Using Cramer’s Rule to Solve a Linear System in Three Variables Use Cramer’s Rule to solve the system: Step 1 Set up the determinants.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Using Cramer’s Rule to Solve a Linear System in Three Variables Use Cramer’s Rule to solve the system: Step 2 Evaluate the four determinants.

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Using Cramer’s Rule to Solve a Linear System in Three Variables Use Cramer’s Rule to solve the system: Step 2 (cont) Evaluate the four determinants.

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Example: Using Cramer’s Rule to Solve a Linear System in Three Variables Use Cramer’s Rule to solve the system: Step 2 (cont) Evaluate the four determinants.

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Using Cramer’s Rule to Solve a Linear System in Three Variables Use Cramer’s Rule to solve the system: Step 2 (cont) Evaluate the four determinants.

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Using Cramer’s Rule to Solve a Linear System in Three Variables Use Cramer’s Rule to solve the system: Step 3 Substitute these four values and solve the system. The solution set is {(2, –3, 4)}.

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 The Determinant of Any n x n Matrix The determinant of a matrix with n rows and n columns is said to be an nth-order determinant. The value of an nth-order determinant (n > 2) can be found in terms of determinants of order n – 1. The minor of the element a ij is the determinant obtained by deleting the ith row and the jth column in the given array of numbers. The cofactor of the element a ij is (– 1) i + j times the minor of a ij.

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Evaluating the Determinant of a 4 x 4 Matrix Evaluate the determinant of the following matrix: With three 0’s in the third column, we will expand along the third column.

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Evaluating the Determinant of a 4 x 4 Matrix (continued) We are evaluating the determinant