1. Precalculus – MAT 129 Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF

2. Chapter Seven Linear Systems and Matrices

3. Ch. 7 Overview Solving Systems of Equations Systems of Linear Equations in Two Variables Multivariable Linear Systems Matrices and Systems of Equations

4. Ch. 7 Overview (cont.) Operations with Matrices The Inverse of a Square Matrix The Determinant of a Square Matrix Applications of Matrices and Determinants

5. 7.1 – Solving Systems of Equations The Method of Substitution

6. 7.1 – The Method of Substitution Systems of equations –the number of equations must equal the number of unknown variables A solution of a system is an ordered pair that satisfies each equation

7. 7.1 – The Method of Substitution 1.Solve one of the equations for one variable. 2.Substitute into the other equation. 3.Solve the equation you substituted into (it will now only be in one variable) 4.Back-substitute into the first equation. 5.Check the solution(s).

8. Example 1.7.1 Pg. 481 #5 Solve the system by the method of substitution. 2x + y = 6 -x + y = 0

9. Solution Example 1.7.1 Pg. 481 #5 Following the steps: Solving for y in the second equation gives y=x. Substituting in the first equation gives 2x + x = 6  3x = 6  x = 2. Back substituting gives y = 2. Check (2,2) in both equations.

10. Things that can happen 1.No-Solution Case The graphs don’t touch. 2.Two-Solution Case The graphs touch in two places.

11. Example 2.7.1 Solve the system by the method of substitution. 6x + 5y = -3 -x – (5/6)y = -7

12. Solution Example 2.7.1 Solving for y in the first equation yields: y = -(3/5) – (6/5)x Substituting for y in the second equation: -x – (5/6)(-3/5 – (6/5)x )  ½ = -7. This means that there is no solution.

13. Example 3.7.1 Pg. 482 #27 Solve the system by the method of substitution. x 3 – y = 0 x – y = 0

14. Solution Example 3.7.1 Pg. 482 #27 Solving for y in the second equation yields: y = x Substituting for y in the first equation: x 3 – x = 0 x(x 2 – 1) = 0 x = 1, -1, and 0. This means that there are multiple solutions.

15. Example 4.7.1 Pg. 478 Example 5 Solve the system of equations graphically. y = ln(x) x + y = 1

16. Solution Example 4.7.1 Looking at the graph of these two equations shows us that the point of intersection is (1,0). Make sure to still check graphical solutions in the equations!

17. Activities (479) 1. Solve the system by substitution. 3x + 2y = 14 and x – 2y = 10. 2. Find all points of intersection. 4x – y – 5 = 0 and 4x 2 – 8x + y + 5 = 0 3. Solve the system graphically. 3x + 2y = 6 and y = ln(x-1)

18. 7.2 – Systems of Linear Equations in Two Variables The Method of Elimination Graphical Interpretation of Two-Variable Systems Application

19. 7.2 – The Method of Elimination 1.Find coefficients that when added will cancel each other out. Remember you can multiply to make this happen! 2.Add the equations to eliminate one variable; solve the resulting equation. 3.Back-substitute into the first equation. 4.Check your solutions in both original equations.

20. Example 1.7.2 Pg. 486 Example 2 Solve the system of linear equations. 5x + 3y = 9 2x - 4y = 14

21. Solution Example 1.7.2 Pg. 486 Example 2 (3, -2)

22. Example 2.7.2 Pg. 491 #11 Solve the system of linear equations. 3r + 2s = 10 2r + 5s = 3

23. Solution Example 2.7.2 Pg. 491 #11 Multiplying: -2*(3r + 2s = 10) and 3*(2r + 5s = 3) Adding: 11s = -11 s = -1

24. Solution Example 2.7.2 (cont.) Substituting: 3r + 2(-1) = 10 3r = 12 r = 4 Checking r = 4 and s = -1 into both original solutions: 12 – 2 = 10 √ 8 – 5 = 3 √

25. 7.2 – Graphical Interpretation of Two-Variable Systems There are three kinds of systems that you will encounter: 1.One solution 2.Infinitely many solutions (identical lines) Equations with at least one solution are called consistent. 3.No solution (parallel lines) These equations are inconsistent.

26. Example 3.7.2 Pg. 487 Example 3 The graphs of the three different types of systems.

27. Example 4.7.2 Pg. 491 #21 Solve the system of linear equations by elimination. 4x + 3y = 3 3x + 11y = 13

28. Solution Example 4.7.2 Pg. 491 #21 Multiplying: -11*(4x + 3y = 3) and 3*(3x + 11y = 13) Adding: -35x = 6 x = -(6/35)

29. Solution Example 4.7.2 (cont.) Substituting: y = 43/35 Checking the solution shows that they work in both equations.

30. 7.3 – Multivariable Linear Systems Row-Echelon Form and Back Substitution Gaussian Elimination Nonsquare Systems Graphical Interpretation of Three-Variable Systems Partial Fraction Decomposition and Other Applications

31. 7.3 – Row-Echelon Form and Back Substitution Row-echelon form is a “stair-step” pattern and all leading coefficients are 1. See Example 1 on pg. 495

32. Example 1.7.3 Pg. 505 #5 Use back-substitution to solve the system of linear equations. 2x – y + 5z = 16 y + 2z = 2 z = 2

33. Solution Example 1.7.3 Pg. 505 #5 Substitute z=6 into the second equation: y + 2(6) = 4 y = -8 Then substitute both z and y into the first: 2x + 8 + 30 = 24 2x = -14 x = -7

34. Solution Example 1.7.3 (cont.) Pg. 505 #5 So the ordered triple that is the solution of this system of equations is: (-7, -8, 6) Check these solutions!

35. 7.3 – Gaussian Elimination Elementary Row Operations 1.Interchange two equations. 2.Multiply one of the equations by a nonzero constant. 3.Add a multiple of one equation to another equation.

36. 7.3 – Gaussian Elimination Use elementary row operations to get the equations in row-echelon form.

37. Example 2.7.3 Pg. 496 Example 2 Read the row operations that were used in the red boxes to the right of the equations.

38. Example 3.7.3 Pg. 506 #15 Solve the system of equations and check any solution. x + y + z = 6 2x – y + z = 3 3x - z = 0 Do on the board.

39. Solution Example 3.7.3 Pg. 506 #15 So the ordered triple that is the solution of this system of equations is: (1, 2, 3) Don’t forget to check these solutions.

40. Example 4.7.3 Pg. 497-498 Examples 3 and 4 Recall these special cases. The same rules apply as when we are in two variables.

41. 7.3 – Nonsquare Systems In a non-square system of equations, the number of equations differs from the number of variables.

42. Example 5.7.3 Pg. 499 Example 5 Pay close attention these are pretty hard.

43. Example 6.7.3 Pg. 506 #37 Solve the system of linear equations and check your solutions! x – 2y + 5z = 2 4x – z = 0

44. Solution Example 6.7.3 Pg. 506 #37 So the ordered triple that is the solution of this system of equations is: (2a, 21a – 1, 8a) Don’t forget to check these solutions!

45. 7.3 – Graphs in Three Dimensions Note the pictures of what you are finding for solutions on the bottom of pg. 500.

46. 7.3 – Partial Fractions Distinct Linear Factors Repeated Linear Factors

47. Example 7.7.3 Pg. 502 Example 6

48. Example 8.7.3 Now look at Example 7 on pg. 503 Note: When you get double roots you have to include the single factor and the squared factor.

49. Activities (499 & 502) 1. Solve the system: x – y + z = 4 x +3y – 2z = -3 3x + 2y + z = 5 2. Solve the system: x – 2y – z = -5 2x + y + z = 5 3 and 4 are partial fractions that I will put on the board.

50. 7.4 – Matrices and Systems of Equations Matrices Elementary Row Operations Gaussian Elimination with Back Substitution Gauss-Jordan Elimination

51. 7.4 – Matrices An m by n matrix is a rectangular array with m rows and n columns. Look at the examples of systems, augmented matrices, and coefficient matrices and be familiar with these terms.

52. 7.4 – Row Operations You can use the exact same row operations as we used in the last section when dealing with matrices.

53. 7.4 – Gaussian Elimination with Back-Substitution 1.Write the augmented matrix of the system of linear equations. 2.Use elementary row operations to rewrite the augmented matrix in row-echelon form. 3.Write the system of linear equations corresponding to the matrix in row- echelon form and use back-substitution to find the solution.

54. Example 1.7.4 Pg. 516 Example 6 Make sure you can follow all of the steps.

55. 7.4 – Gauss-Jordan Elimination Using this technique you don’t stop until you have reached reduced row-echelon form with your matrix. Then the numbers on the other side of the line in the augmented matrix are your solutions. Reduced means zeros above and below the diagonals of ones.

56. Example 2.7.4 Pg. 523 #49 x + 2y = 7 2x + y = 8

57. Solution Example 2.7.4 Pg. 523 #49 1 0 | 3 0 1 | 2 So this gives an ordered pair: (3,2) Make sure you check this solution.

58. Example 3.7.4 Pg. 523 #55 x - 3z = -2 3x + y – 2z = 5 2x + 2y + z = 4

59. Solution Example 3.7.4 Pg. 523 #55 (4, -3, 2) Make sure you check this solution.

60. Activity (519) Use Gauss-Jordan elimination to solve the system (show your steps): x + 2y + z = -4 2x - y + z = -4 x + 3y - z = -7

61. 7.5 – Operations with Matrices Equality of Matrices Matrix Addition and Scalar Multiplication Matrix Multiplication Applications

62. 7.5 – Equality of Matrices A matrix is equal to another if the dimensions are the same and the entries are all the same.

63. 7.5 – Addition of Matrices Matrix addition is a piece-wise addition and therefore the dimensions need to be exactly the same. Your calculator will do this for you!! Make sure you know how to enter them and do operations with them. We will do an example.

64. Example 1.7.5 Pg. 527 Example 2 Do this in your calculator.

65. 7.5 – Scalar Multiplication of Matrices This also is a piece-wise action and the scalar is simply multiplied by each entry in the matrix. The product is a matrix of the same size as the original one. Your calculator will do this for you!! Make sure you know how to enter them and do operations with them.

66. Example 2.7.5 Pg. 529 Example 4 Do this in your calculator. Note the properties of addition and scalar multiplication in the blue box on pg. 529.

67. 7.5 – Matrix Multiplication This function is not a piecewise operation. The definition is on pg. 509 in the blue box. We will usually use our calculators to do this for us. Do Example 8 on pg. 509 by hand.

68. Example 3.7.5 Pg. 533 Example 10 Do this in your calculator. Note the properties of multiplication in the blue box on pg. 511.

69. 7.5 – The Identity Matrix This is a square matrix in which all of the diagonal entries are ones and all of the off-diagonal entries are zero. See the definition on pg. 533.

70. 7.6 – The Inverse of a Square Matrix The Inverse of a Matrix Finding Inverse Matrices The Inverse of a 2X2 Matrix Systems of Linear Equations

71. 7.6 – The Inverse of a Matrix We denote the inverse of a matrix A by A -1 (pronounced A inverse). AA -1 = I = A -1 A Where I is the identity matrix of the same size as A. Note that A is a square matrix.

72. 7.6 – Finding the Inverse of a Matrix Example 3 on pg. 544 shows how to do this by hand.

73. Example 1.7.6 Pg. 545 Example 4 Do this in your calculator.

74. 7.6 – The Inverse of a 2x2 Matrix There is a formula that you can use to easily find the inverse of a 2x2 matrix. This is nicely laid out on pg. 545 of your text.

75. Example 2.7.6 Pg. 545 Example 5 Find the inverse using the formula.

76. 7.6 – Systems of Linear Equations If A has in inverse: Ax = B has the unique solution x = A -1 B.

77. Example 3.7.6 Pg. 546 Example 6 If you understand how this works solving these systems on your calculator should be easy.

78. 7.7 – The Determinant of a Square Matrix The Determinant of a 2X2 Matrix Minors and Cofactors The Determinant of a Square Matrix Triangular Matrices

79. 7.7 – The Determinant of a 2X2 Matrix The determinant of the matrix A = a1 b1 a2 b2 is given by det(A) = |A| = a1b2 – a2b1.

80. Example 1.7.7 Pg. 552 Example 1 Do this by hand and by calculator.

81. 7.7 – Minors and Cofactors The minors and cofactors come in when we are dealing with square matrices of size 3x3 or bigger. Each entry in the matrix has a minor and a cofactor. We use these to find the determinants of square matrices.

82. 7.7 – Minors and Cofactors To find a minor you cross out the row and column of the entry in question and find the determinant of the matrix remaining. A minor becomes a cofactor when it is either multiplied by 1 or -1 depending on the place of the original entry.

83. Example 2.7.7 Pg. 553 Example 2 If you remember that the first entry is positive and it alternates in both directions throughout the whole matrix you shouldn’t need the charts given on pg. 553.

84. 7.7 – The Determinant of a Square Matrix You can find the determinant of any square matrix by adding the products of the entries of any row or column and their cofactors.

85. Example 3.7.7 Pg. 556 #23 There is also another way that works for just 3x3 matrices. I will show you both the cofactor and the trick way to find the determinant of this matrix.

86. 7.7 – Triangular Matrices If you have either an upper-triangular, lower- triangular or a diagonal matrix there is a really easy way to find the determinant. Simply multiply the entries on the diagonal.

87. Example 4.7.7 Pg. 555 Example 4 The hardest part about these is noticing that they are the right kind to use the trick. So, the key is to make sure you understand what a triangular matrix looks like in any of the three forms.

88. 7.8 – Applications of Matrices and Determinants Area of a Triangle Collinear Points Cramer’s Rule Cryptography