1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 7 Systems and Matrices

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.1 Solving Systems of Two Equations

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 What you’ll learn about The Method of Substitution Solving Systems Graphically The Method of Elimination Applications … and why Many applications in business and science can be modeled using systems of equations.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 Solution of a System A solution of a system of two equations in two variables is an ordered pair of real numbers that is a solution of each equation.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Example Using the Substitution Method

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Example Using the Substitution Method

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 10 Example Solving a Nonlinear System Algebraically

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 11 Example Solving a Nonlinear System Algebraically

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 12 Example Using the Elimination Method

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 13 Example Using the Elimination Method

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 14 Example Finding No Solution

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 15 Example Finding No Solution

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 16 Example Finding Infinitely Many Solutions

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 17 Example Finding Infinitely Many Solutions

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.2 Matrix Algebra

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 19 Quick Review

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 20 Quick Review Solutions

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 21 What you’ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 22 Matrix

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 23 Matrix Vocabulary Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is the ith row and the jth column. In general, the order of an m × n matrix is m×n.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 24 Example Determining the Order of a Matrix

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 25 Example Determining the Order of a Matrix

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 26 Matrix Addition and Matrix Subtraction

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 27 Example Matrix Addition

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 28 Example Matrix Addition

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 29 Example Using Scalar Multiplication

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 30 Example Using Scalar Multiplication

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 31 The Zero Matrix

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 32 Additive Inverse

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 33 Matrix Multiplication

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 34 Example Matrix Multiplication

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 35 Example Matrix Multiplication

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 36 Identity Matrix

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 37 Inverse of a Square Matrix

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 38 Inverse of a 2 × 2 Matrix

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 39 Determinant of a Square Matrix

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 40 Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0.

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 41 Example Finding Inverse Matrices

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 42 Example Finding Inverse Matrices

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 43 Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Community property Addition: A + B = B + A Multiplication: Does not hold in general 2. Associative property Addition: (A + B) + C = A + (B + C) Multiplication: (AB)C = A(BC) 3. Identity property Addition: A + 0 = A Multiplication: A·I n = I n ·A = A 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA -1 = A -1 A = I n |A|≠0 5. Distributive property Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.3 Multivariate Linear Systems and Row Operations

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 45 Quick Review

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 46 Quick Review Solutions

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 47 What you’ll learn about Triangular Forms for Linear Systems Gaussian Elimination Elementary Row Operations and Row Echelon Form Reduced Row Echelon Form Solving Systems with Inverse Matrices Applications … and why Many applications in business and science are modeled by systems of linear equations in three or more variables.

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 48 Equivalent Systems of Linear Equations The following operations produce an equivalent system of linear equations. 1. Interchange any two equations of the system. 2. Multiply (or divide) one of the equations by any nonzero real number. 3. Add a multiple of one equation to any other equation in the system.

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 49 Row Echelon Form of a Matrix A matrix is in row echelon form if the following conditions are satisfied. 1. Rows consisting entirely of 0’s (if there are any) occur at the bottom of the matrix. 2. The first entry in any row with nonzero entries is 1. 3. The column subscript of the leading 1 entries increases as the row subscript increases.

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 50 Elementary Row Operations on a Matrix A combination of the following operations will transform a matrix to row echelon form. 1. Interchange any two rows. 2. Multiply all elements of a row by a nonzero real number. 3. Add a multiple of one row to any other row.

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 51 Example Finding a Row Echelon Form

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 52 Example Finding a Row Echelon Form

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 53 Reduced Row Echelon Form If we continue to apply elementary row operations to a row echelon form of a matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the reduced echelon form.

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 54 Example Solving a System Using Inverse Matrices

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 55 Example Solving a System Using Inverse Matrices

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.4 Partial Fractions

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 57 Quick Review

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 58 Quick Review Solutions

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 59 What you’ll learn about Partial Fraction Decomposition Denominators with Linear Factors Denominators with Irreducible Quadratic Factors Applications … and why Partial fraction decompositions are used in calculus in integration and can be used to guide the sketch of the graph of a rational function.

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 60 Partial Fraction Decomposition of f(x)/d(x)

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 61 Example Decomposing a Fraction with Distinct Linear Factors

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 62 Example Decomposing a Fraction with Distinct Linear Factors

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 63 Example Decomposing a Fraction with an Irreducible Quadratic Factor

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 64 Example Decomposing a Fraction with an Irreducible Quadratic Factor

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.5 Systems of Inequalities in Two Variables

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 66 Quick Review

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 67 Quick Review Solutions

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 68 What you’ll learn about Graph of an Inequality Systems of Inequalities Linear Programming … and why Linear programming is used in business and industry to maximize profits, minimize costs, and to help management make decisions.

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 69 Steps for Drawing the Graph of an Inequality in Two Variables 1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is. Use a solid line if the inequality is ≤ or ≥. 2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 70 Example Graphing a Linear Inequality

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 71 Example Graphing a Linear Inequality

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 72 Example Solving a System of Inequalities Graphically

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 73 Example Solving a System of Inequalities Graphically

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 74 Chapter Test

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 75 Chapter Test

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 76 Chapter Test

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 77 Chapter Test

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 78 Chapter Test Solutions

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 79 Chapter Test Solutions

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 80 Chapter Test Solutions

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 81 Chapter Test Solutions