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Int Math 2 Section 8-5

J
Jimbo Lamb

The document provides information about matrices and determinants. It defines key terms like square matrix, determinant, and Cramer's rule. It gives examples of how to find the determinant of a 2x2 matrix and how to use Cramer's rule to solve systems of equations. The examples show setting up the matrices, calculating the determinants, and solving for the variables.

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SECTION 8-5 Matrices and Determinants Tue, Apr 12
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, fitness Tue, Apr 12
VOCABULARY 1. Square Matrix: 2. Determinant: 3. Cramer’s Rule: Tue, Apr 12
VOCABULARY 1. Square Matrix: A matrix with the same number of rows and columns 2. Determinant: 3. Cramer’s Rule: Tue, Apr 12
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
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 equations Tue, Apr 12
MATRIX m x n: m rows and n columns Tue, Apr 12
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
MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3 Tue, Apr 12
MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3 2X2 Tue, Apr 12
MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3 2X2 1X3 Tue, Apr 12
DETERMINANT a b det A = = ad − bc c d Tue, Apr 12
EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ Tue, Apr 12
EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc Tue, Apr 12
EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) Tue, Apr 12
EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) = 0 − 24 Tue, Apr 12
EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) = 0 − 24 = −24 Tue, Apr 12
CRAMER’S RULE Tue, Apr 12
CRAMER’S RULE 1. Make sure equations look like Ax + By = C. Tue, Apr 12
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
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
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
CRAMER’S RULE Tue, Apr 12
CRAMER’S RULE 5. Divide Ax by A and Ay by A. Tue, Apr 12
CRAMER’S RULE 5. Divide Ax by A and Ay by A. 6. Check answer and rewrite solution. Tue, Apr 12
EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 ⎨ ⎩ x + 2 y = 11 Tue, Apr 12
EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 3 −7 ⎨ A= ⎩ x + 2 y = 11 1 2 Tue, Apr 12
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 2 Tue, Apr 12
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 Tue, Apr 12
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= A Tue, Apr 12
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
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+7 Tue, Apr 12
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 13 Tue, Apr 12
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 Tue, Apr 12
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= A Tue, Apr 12
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
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+7 Tue, Apr 12
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 13 Tue, Apr 12
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 13 Tue, Apr 12
EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Tue, Apr 12
EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: Tue, Apr 12
EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 Tue, Apr 12
EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 15 − 21 = −6 Tue, Apr 12
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 Tue, Apr 12
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 Tue, Apr 12
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
PROBLEM SET Tue, Apr 12
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 Jefferson Tue, Apr 12

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Int Math 2 Section 8-5

  • 1. SECTION 8-5 Matrices and Determinants Tue, Apr 12
  • 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, fitness Tue, Apr 12
  • 3. VOCABULARY 1. Square Matrix: 2. Determinant: 3. Cramer’s Rule: Tue, Apr 12
  • 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. 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. 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 equations Tue, Apr 12
  • 7. MATRIX m x n: m rows and n columns Tue, Apr 12
  • 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. MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3 Tue, Apr 12
  • 10. MATRIX m x n: m rows and n columns ⎡ −2 2 3 ⎤ ⎡3 −1⎤ ⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤ ⎣ ⎦ ⎣ 9 6 −3 ⎦ ⎣5 8 ⎦ 2X3 2X2 Tue, Apr 12
  • 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 1X3 Tue, Apr 12
  • 12. DETERMINANT a b det A = = ad − bc c d Tue, Apr 12
  • 13. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ Tue, Apr 12
  • 14. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc Tue, Apr 12
  • 15. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) Tue, Apr 12
  • 16. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) = 0 − 24 Tue, Apr 12
  • 17. EXAMPLE 1 ⎡0 4 ⎤ Find the determinant of ⎢ ⎥. ⎣6 7 ⎦ ad − bc = 0(7) − 4(6) = 0 − 24 = −24 Tue, Apr 12
  • 19. CRAMER’S RULE 1. Make sure equations look like Ax + By = C. Tue, Apr 12
  • 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. 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. 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
  • 24. CRAMER’S RULE 5. Divide Ax by A and Ay by A. Tue, Apr 12
  • 25. CRAMER’S RULE 5. Divide Ax by A and Ay by A. 6. Check answer and rewrite solution. Tue, Apr 12
  • 26. EXAMPLE 2 Solve the system of equations using Cramer’s rule (matrices). ⎧3x − 7 y = −6 ⎨ ⎩ x + 2 y = 11 Tue, Apr 12
  • 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 2 Tue, Apr 12
  • 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 2 Tue, Apr 12
  • 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 11 Tue, Apr 12
  • 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= A Tue, Apr 12
  • 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. 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+7 Tue, Apr 12
  • 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 13 Tue, Apr 12
  • 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 13 Tue, Apr 12
  • 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= A Tue, Apr 12
  • 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. 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+7 Tue, Apr 12
  • 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 13 Tue, Apr 12
  • 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 13 Tue, Apr 12
  • 40. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Tue, Apr 12
  • 41. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: Tue, Apr 12
  • 42. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 Tue, Apr 12
  • 43. EXAMPLE 2 ⎧3x − 7 y = −6 ⎨ x = 5, y = 3 ⎩ x + 2 y = 11 Check: 3(5) − 7(3) = −6 15 − 21 = −6 Tue, Apr 12
  • 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 = −6 Tue, Apr 12
  • 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 = 11 Tue, Apr 12
  • 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
  • 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 Jefferson Tue, Apr 12