1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 9 Matrices and Determinants

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Determinants and Cramer’s Rule Calculate the determinant of a 2 × 2 matrix. Find minors and cofactors. Evaluate the determinant of an n × n matrix. Apply Cramer’s Rule. SECTION

3 © 2010 Pearson Education, Inc. All rights reserved DETERMINANT OF A 2 × 2 MATRIX The determinant of the matrix denoted by det(A), |A| or is called a 2 by 2 determinant and is defined by

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Calculating the Determinant of a 2 × 2 Matrix Evaluate each determinant. Solution

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Calculating the Determinant of a 2 × 2 Matrix Solution continued

6 © 2010 Pearson Education, Inc. All rights reserved MINORS AND COFACTORS IN AN n × n MATRIX Let A be an n × n square matrix. The minor M ij of the element a ij is the determinant of the (n – 1) × (n – 1) matrix obtained by deleting the ith row and the jth column of A. The cofactor of A ij of the entry a ij is given by:

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding Minors and Cofactors For the matrix find: Solution a. the minors M 11, M 23, and M 32 b. the cofactors A 11, A 23, and A 32 a. (i) To find M 11 delete the first row and first column of the matrix A.

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding Minors and Cofactors Solution continued Now find the determinant of the resulting matrix. (ii) To find M 23 delete the second row and third column of the matrix A.

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding Minors and Cofactors Solution continued Now find the determinant of the resulting matrix. (iii) To find M 32 delete the third row and second column of the matrix A.

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding Minors and Cofactors Solution continued So b. To find the cofactors, use the formula A ij = (−1) i+j M ij

11 © 2010 Pearson Education, Inc. All rights reserved n × n DETERMINANT Let A be a square matrix of order n ≥ 3. The determinant of A is the sum of the entries in any row of A (or column of A), multiplied by their respective cofactors.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Evaluating a 3 by 3 Determinant Find the determinant of Solution Expanding by the first row, we have where

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Evaluating a 3 by 3 Determinant Solution continued Thus

14 © 2010 Pearson Education, Inc. All rights reserved CRAMER’S RULE FOR SOLVING TWO EQUATIONS IN TWO VARIABLES The system provided that D ≠ 0, where two variables has a unique solution (x, y) given by of two equations in

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Using Cramer’s Rule Use Cramer’s rule to solve the system: Solution Step 1Form the determinant D of the coefficient matrix. Since D = 7 ≠ 0, the system has a unique solution.

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Using Cramer’s Rule Solution continued Step 2Replace the column of coefficients of x in D by the constant terms to obtain Step 3Replace the column of coefficients of y in D by the constant terms to obtain

17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Using Cramer’s Rule The solution of the system is x = –3 and y = 4. Solution continued Step 4By Cramer’s rule, The solution set is {(–3, 4)}. Check:You should check the solution in the original system.

18 © 2010 Pearson Education, Inc. All rights reserved CRAMER’S RULE FOR SOLVING THREE EQUATIONS IN THREE VARIABLES The system provided that D ≠ 0, where, equations in three variables has a unique solution (x, y, z) given by of three

19 © 2010 Pearson Education, Inc. All rights reserved CRAMER’S RULE FOR SOLVING THREE EQUATIONS IN THREE VARIABLES (continued)

20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using Cramer’s Rule Use Cramer’s rule to solve the system of equations. Solution Rewrite the system

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using Cramer’s Rule Solution continued Step 1 D ≠ 0, so the system has a unique solution.

22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using Cramer’s Rule Solution continued Step 2Replace first column in D by constants.

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using Cramer’s Rule Solution continued Step 3Replace 2 nd column in D by constants.

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using Cramer’s Rule Solution continued Step 4Replace 3rd column in D by constants.

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using Cramer’s Rule Solution continued Step 5Cramer’s rule gives the following values. Hence, the solution set is {(–1, 3, 5)}. Check: You should check the solution.