1. 5.3 - 1 Determinants Every n  n matrix A is associated with a real number called the determinant of A, written  A . The determinant of a 2  2 matrix is defined as follows.

2. 5.3 - 2 Determinant of a 2  2 Matrix If A =

3. 5.3 - 3 Note Matrices are enclosed with square brackets, while determinants are denoted with vertical bars. A matrix is an array of numbers, but its determinant is a single number.

4. 5.3 - 4 Determinants The arrows in the following diagram will remind you which products to find when evaluating a 2  2 determinant.

5. 5.3 - 5 Example 1 EVALUATING A 2  2 DETERMINANT Let A = Find  A . Use the definition with Solution a 11 a 22 a 21 a 12

6. 5.3 - 6 Determinant of a 3  3 Matrix If A =

7. 5.3 - 7 Evaluating The terms on the right side of the equation in the definition of  A  can be rearranged to get Each quantity in parentheses represents the determinant of a 2  2 matrix that is the part of the matrix remaining when the row and column of the multiplier are eliminated, as shown in the next slide.

8. 5.3 - 8 Evaluating

9. 5.3 - 9 Cramer’s Rule for Two Equations in Two Variables Given the system if then the system has the unique solution where

10. 5.3 - 10 Caution As indicated in the preceding box, Cramer’s rule does not apply if D = 0. When D = 0 the system is inconsistent or has infinitely many solutions. For this reason, evaluate D first.

11. 5.3 - 11 Example 4 APPLYING CRAMER’S RULE TO A 2  2 SYSTEM Use Cramer’s rule to solve the system Solution By Cramer’s rule, and Find D first, since if D = 0, Cramer’s rule does not apply. If D ≠ 0, then find D x and D y.

12. 5.3 - 12 Example 4 APPLYING CRAMER’S RULE TO A 2  2 SYSTEM By Cramer’s rule, The solution set is as can be verified by substituting in the given system.

13. 5.3 - 13 General form of Cramer’s Rule Let an n  n system have linear equations of the form Define D as the determinant of the n  n matrix of all coefficients of the variables. Define D x1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define D xi as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D  0, the unique solution of the system is

14. 5.3 - 14 Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Use Cramer’s rule to solve the system. Solution Rewrite each equation in the form ax + by + cz +  = k.

15. 5.3 - 15 Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Verify that the required determinants are

16. 5.3 - 16 Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Thus, and so the solution set is

17. 5.3 - 17 Note When D = 0, the system is either inconsistent or has infinitely many solutions. Use the elimination method to tell which is the case. Verify that the system in Example 6 is inconsistent, so the solution set is ø.