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# Int Math 2 Section 8-5

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Matrices and Determinants

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### Int Math 2 Section 8-5

1. 1. SECTION 8-5 Matrices and DeterminantsTue, Apr 12
2. 2. ESSENTIAL QUESTIONS How do you ﬁnd the determinant of a 2X2 matrix? How do you solve systems of equations using determinants? Where you’ll see this: Sports, construction, ﬁtnessTue, Apr 12
3. 3. VOCABULARY 1. Square Matrix: 2. Determinant: 3. Cramer’s Rule:Tue, Apr 12
4. 4. VOCABULARY 1. Square Matrix: A matrix with the same number of rows and columns 2. Determinant: 3. Cramer’s Rule:Tue, Apr 12
5. 5. VOCABULARY 1. Square Matrix: A matrix with the same number of rows and columns ⎡a b ⎤ 2. Determinant: In a 2X2 matrix ⎢ ⎥ , det A = ad - bc. ⎣c d ⎦ 3. Cramer’s Rule:Tue, Apr 12
6. 6. VOCABULARY 1. Square Matrix: A matrix with the same number of rows and columns ⎡a b ⎤ 2. Determinant: In a 2X2 matrix ⎢ ⎥ , det A = ad - bc. ⎣c d ⎦ 3. Cramer’s Rule: A method of using determinants of matrices to solve systems of equationsTue, Apr 12
7. 7. MATRIX m x n: m rows and n columnsTue, Apr 12
8. 8. MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦Tue, Apr 12
9. 9. MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3Tue, Apr 12
10. 10. MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3 2X2Tue, Apr 12
11. 11. MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3 2X2 1X3Tue, Apr 12
12. 12. DETERMINANT a b det A = = ad − bc c dTue, Apr 12
13. 13. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦Tue, Apr 12
14. 14. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bcTue, Apr 12
15. 15. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6)Tue, Apr 12
16. 16. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) = 0 − 24Tue, Apr 12
17. 17. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) = 0 − 24 = −24Tue, Apr 12
18. 18. CRAMER’S RULETue, Apr 12
19. 19. CRAMER’S RULE 1. Make sure equations look like Ax + By = C.Tue, Apr 12
20. 20. CRAMER’S RULE 1. Make sure equations look like Ax + By = C. 2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A.Tue, Apr 12
21. 21. CRAMER’S RULE 1. Make sure equations look like Ax + By = C. 2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A. 3. Make a new 2X2 determinant matrix: Replace x column with equation answers, call Ax.Tue, Apr 12
22. 22. CRAMER’S RULE 1. Make sure equations look like Ax + By = C. 2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A. 3. Make a new 2X2 determinant matrix: Replace x column with equation answers, call Ax. 4. Make another 2X2 determinant matrix: Replace y column with equation answers, call Ay.Tue, Apr 12
23. 23. CRAMER’S RULETue, Apr 12
24. 24. CRAMER’S RULE 5. Divide Ax by A and Ay by A.Tue, Apr 12
25. 25. CRAMER’S RULE 5. Divide Ax by A and Ay by A. 6. Check answer and rewrite solution.Tue, Apr 12
26. 26. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 ⎨ ⎩ x + 2 y = 11Tue, Apr 12
27. 27. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 ⎨ A= ⎩ x + 2 y = 11 1 2Tue, Apr 12
28. 28. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 ⎨ A= Ax = ⎩ x + 2 y = 11 1 2 11 2Tue, Apr 12
29. 29. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11Tue, Apr 12
30. 30. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax x= ATue, Apr 12
31. 31. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) x= = A (3)(2) − (−7)(1)Tue, Apr 12
32. 32. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 x= = = A (3)(2) − (−7)(1) 6+7Tue, Apr 12
33. 33. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 65 x= = = = A (3)(2) − (−7)(1) 6+7 13Tue, Apr 12
34. 34. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 65 x= = (3)(2) − (−7)(1) = 6+7 = =5 A 13Tue, Apr 12
35. 35. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 65 x= = (3)(2) − (−7)(1) = 6+7 = =5 A 13 Ay y= ATue, Apr 12
36. 36. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 65 x= = (3)(2) − (−7)(1) = 6+7 = =5 A 13 Ay (3)(11) − (−6)(1) y= = A (3)(2) − (−7)(1)Tue, Apr 12
37. 37. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 65 x= = (3)(2) − (−7)(1) = 6+7 = =5 A 13 (3)(11) − (−6)(1) 33 + 6 Ay y= = = A (3)(2) − (−7)(1) 6+7Tue, Apr 12
38. 38. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 65 x= = (3)(2) − (−7)(1) = 6+7 = =5 A 13 (3)(11) − (−6)(1) 33 + 6 39 Ay y= = = = A (3)(2) − (−7)(1) 6 + 7 13Tue, Apr 12
39. 39. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 −6 −7 3 −6 ⎨ A= Ax = Ay = ⎩ x + 2 y = 11 1 2 11 2 1 11 Ax(−6)(2) − (−7)(11) −12 + 77 65 x= = (3)(2) − (−7)(1) = 6+7 = =5 A 13 (3)(11) − (−6)(1) 33 + 6 39 Ay y= = = = =3 A (3)(2) − (−7)(1) 6 + 7 13Tue, Apr 12
40. 40. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11Tue, Apr 12
41. 41. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check:Tue, Apr 12
42. 42. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6Tue, Apr 12
43. 43. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 15 − 21 = −6Tue, Apr 12
44. 44. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 5 + 2(3) = 11 15 − 21 = −6Tue, Apr 12
45. 45. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 5 + 2(3) = 11 15 − 21 = −6 5 + 6 = 11Tue, Apr 12
46. 46. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 5 + 2(3) = 11 15 − 21 = −6 5 + 6 = 11 (5,3)Tue, Apr 12
47. 47. PROBLEM SETTue, Apr 12
48. 48. PROBLEM SET p. 356 #1-31 odd Solve all using matrices by hand “I’m a great believer in luck, and I ﬁnd the harder I work the more I have of it.” - Thomas JeffersonTue, Apr 12