1. Chapter 4 Matrices

2. In Chapter 4, You Will… Move from using matrices in organizing data to manipulating matrices through data. Learn to represent real-world relationships by writing matrices and using operations such as addition and multiplication to develop new matrices.

3. 4-1 Organizing Data What you’ll learn … To identify matrices and their elements To organize data into matrices 1.04 Operate with matrices to model and solve problems.

4.  A matrix (plural matrices) is a rectangular array of numbers written within brackets.  The number of horizontal rows and the number of vertical columns determine the dimensions of a matrix. Rows Columns

5. Example 1 Writing the Dimensions of a Matrix Write the dimensions of each matrix.

6. Each number in a matrix is a matrix element. You can identify a matrix element by its position within the matrix. Use a lowercase letter with subscripts. The subscripts represent the element’s row number and column number. Consider the matrix The element a 21 = 1, since the element in the 2nd row and 1st column is 1. The element a 13 = 9, since the element in the 1st row and 3rd column is 9.

7. Example 2 Identifying a Matrix Element Identify each matrix element a. a 33 b. a 11 c. a 21 d. a 12 17243 10.41215 93015

8. Example 4a Real World Example Write a matrix W to represent the information. GymnastFloor Exercise VaultBalance Beam Uneven Bars Amy Chow9.5259.4689.6259.400 Dominique Dawes 9.0879.3938.6009.675 Kristin Maloney 9.5259.2259.3129.575 Elise Ray9.2259.4689.687 Which element represents Maloney’s score on the vault?

9. Example 4b Real World Example

10. Example 4b Real World Example Continued Write a matrix M to represent the data in the graph, with columns representing years. What are the dimensions of this matrix? What does the first row represent? What does m 32 represent?

11. 4-2 Adding and Subtracting Matrices What you’ll learn … To add and subtract matrices. To solve certain matrix equations. 1.04 Operate with matrices to model and solve problems.

12. Adding and Subtracting Matrices You must perform matrix addition or subtraction on matrices with equal dimensions by adding or subtracting the corresponding elements, which are elements in the same position in each matrix.

13. Example Adding Matrices 1 -2 03 9 -3 3 -5 7-9 6 12 -12 24 -3 1 -3 5 2 -4 -1 10 -1 5

14. The additive identity matrix for the set of all m x n matrices is the zero matrix 0, or O mxn,whose elements are all zeros. The opposite, or additive inverse, of an m x n matrix A is –A. -A is the m x n matrix with elements that are the opposites of the corresponding elements of A. A + (-A) = 0 A + 0 = A 2-4 -1 0 -2 4 1 0 A =-A =

15. Example 2 Using Identity and Inverse Matrices 1 2 0 0 5 -70 0 2 8 -2 -8 -3 0 3 0

16. Properties of Matrix Addition If A, B, and C are m x n matrices, then A + B is an m x n matrix Closure Property A + B = B + A Commutative Property of Addition (A+B)+C = A+(B+C) Associative Property of Addition There exist a unique m x n matrix O such that O+A=A+O=A.Additive Identity Property For each A, there exists a unique opposite, -A. A+(-A)=0Additive Inverse Property

17. Subtracting Matrices 1 -2 03 9 -3 3 -5 7-9 6 12 1 -2 0-3 -9 3 3 -5 79 -6 -12 Just Add the Opposite

18. Example 3 Subtracting Matrices 6 -9 7-4 3 0 -2 1 86 5 10 -3 5 -3 1 -1 -10 2 -4

19. A matrix equation is an equation in which the variable is a matrix. You can use the addition and subtraction properties of equality to solve matrix equations. Equal matrices are matrices with the same dimensions and equal corresponding elements. X + = -1 0 2 5 10 7 -4 4

20. Example 4 Solving A Matrix Equation Solve X - = Solve X + = 1 3 2 0 1 8 9 -1 0 2 5 107 -4 4

21. Example 5 Determining Equal Matrices -0.75 1/5-3/4 0.2 ½ -2 0.5 -2 4 6 8/2 18/3 16/2 8 Determine whether the two matrices in each pair are equal.

22. Example 6 Finding Unknown Matrix Elements 2x-5 4254 3 3y+123y+18 x+8 -5 38 -5 3 -y 3 4y-10 X = ____ Y = ____ X = ____ Y = ____

23. 4-3 Matrix Multiplication What you’ll learn … To multiply a matrix by a scalar To multiply two matrices 1.04 Operate with matrices to model and solve problems.

24. You can multiply a matrix by a real number. The real number factor (such as 3) is called a scalar. You find the scalar product by multiplying each element of the matrix by the scalar. 3 3 5 2 8 = 9 15 6 24

25. Example Scalar Multiplication 15 -12 10 0 20 -10 7 0 -3

26. Example 2 Using Scalar Products 2 3 -7 1 4 5 A= B= 3 0 6 -1 8 2 Find 5B- 4AFind A + 6B

27. Properties of Scalar Multiplication If A, B, and O are m x n matrices and c and d are scalars, then cA is an m x n matrix Closure property (cd)A = c(dA) Associative Property of Multiplication C(A+B) = cA+cB (c+d)A = cA + cb 1 (A) = A Multiplication Identity Property 0(A) = c0 = 0 Multiplication Property of 0 Distributive Property

28. Example 3a Solving Matrix Equations with Scalars 3 4 -2 1 4x + 2 = 10 0 4 2

29. Example 3b Solving Matrix Equations with Scalars 7 0 -1 2 -3 4 -3x + = 10 0 8 -19 -18 10

30. Investigation: Using Matrices 1.How much money did the cafeteria collect selling lunch 1? Selling Lunch 2? Selling Lunch 3? 2.a. How much did the cafeteria collect selling all 3 lunches? b. Explain how you used the data in the table to find your answer. 3.a. Write a 1x3 matrix to represent the cost of the lunches. b. Write a 1x 3 matrix to represent the number of lunches sold. c. Describe a procedure for using your matrices to find how much money the cafeteria collected from selling all three lunches. Lunch 1 Lunch 2Lunch 3 Cost per Lunch $2.50$1.75$2.00 Number Sold5010075

31. To perform matrix multiplication, multiply the elements of each row of the first matrix by the elements of each column of the second matrix. Add the products.

32. **Multiply rows times columns. **You can only multiply if the number of columns in the 1 st matrix is equal to the number of rows in the 2 nd matrix. Dimensions: 3 x 2 2 x 3 They must match. The dimensions of your answer.

33. Dimensions: 2 x 3 2 x 2 *They don’t match so can’t be multiplied together.* **Multiply rows times columns. **You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix

34. 2 x 2 2 x 2 *Answer should be a 2 x 2 0(4) + (-1)(-2)0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5) 0 -1 1 0 4-3 -2 5

35. Example: -2(4)+5(2)-2(-4)+5(6) 3(4) + -1(2) 3(-4) + -1(6) -2 5 3 -1 4 -4 2 6 2 38 10 -18

36. Example 4a Multiplying Matrices Find the product of Multiply a11 and b11. Add the products. The result is the element in the first row and first column. Repeat with the rest of the rows and columns. -1 0 3 -4 -3 3 5 0

37. Example 4b Multiplying Matrices Find the product of Multiply a11 and b11. Add the products. The result is the element in the first row and first column. Repeat with the rest of the rows and columns. -3 3 5 0 -1 0 3 -4

38. Example 5 Real World Connection A used-record store sells tapes, LP records, and compact discs. The matrices show today’s information. Find the store’s gross income for the day. Tapes LPs CDs Tapes LPs CDs $8 $6 $13 9 30 20

39. Example 5 Find each product. 12 3 10 -5 10 -5 123 0

40. Example 6a Determining When a Product Matrix Exists Use matrices G = and H =. Determine whether products GH and HG are defined (exist) or undefined (do not exist). 80 2 -5 23 -1 8 4 0

41. Example 6b Determining When a Product Matrix Exists Use matrices R = and S =. Determine whether products RS and SR are defined (exist) or undefined (do not exist). 80 -1 0 2 -5 1 8 4-2 5 -4

42. Properties of Matrix Multiplication If A, B, and C are n x n matrices, then AB is an n x n Closure Property (AB)C = A(BC) Associative Property of Multiplication A(B+C) = AB + BC (B+C)A = BA + CA OA = AO = O, where O has the same dimensions as A. Multiplicative Property of 0 Distributive Property

43. 4-5 2x2 Matrices, Determinants, and Inverses What you’ll learn … To evaluate determinants of 2x2 matrices and find inverse matrices To use inverse matrices in solving matrix equations 1.04 Operate with matrices to model and solve problems.

44. Evaluating Determinants of 2x2 Matrices A square matrix is a matrix with the same number of columns as rows. For an n x n square matrix, the multiplicative identity matrix is an n x n square matrix I, with 1’s along the diagonal and zeros elsewhere.

45. Multiplicative Inverse of a Matrix If A and X are n x n matrices, and AX=XA=I, then X is the multiplicative inverse of A, written A -1. A (A -1 ) = A -1 (A) = I

46. Example 1 Verifying Inverses A =B = 2 3 1 2 2 -3 -1 2 M = 3 -1 7 -1 N =.1 -.7.3

47. Determinant of a 2x2 Matrix The determinant of a 2x2 matrix is ad – bc. a b c d -5(-3) – 7(2) 15 – 14 1

48. Example 2 Evaluating the Determinants of a 2x2 Matrix det 4 2 8 7 2 3 det k 3 3-k -3

49. Inverse of a 2x2 Matrix Let A =. If det A ≠0 then A has an inverse. If det A ≠0, then A -1 = a b c d 1__ detA = __1__ ad - bc = d -b -c a

50. Example 3 Finding an Inverse Matrix 2 4 1 3.5 2.3 3 7.2 12 4 9 3 6 5 25 20

51. Example 4 Solving a Matrix Equation - 2 -5 1 3 -2 2 3 -4 4 -5 0 -22 0 -28 X = AX = B A -1 AX = A -1 B IX = A -1 B X= A -1 B

52. 4-6 3x3 Matrices, Determinants, and Inverses What you’ll learn … To evaluate determinants of 3x3 matrices 1.04 Operate with matrices to model and solve problems.

53. Matrix Determinants  A Determinant is a real number associated with a matrix. Only SQUARE matrices have a determinant.  The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A| or.

54. Matrix Determinants To find the determinant of a 3 x 3 matrix, first recopy the first two columns. Then do 6 diagonal products. -20 -24 36 18 6016 Step 1: Rewrite first two columns of the matrix. Step 2: multiply diagonals going up! Step 3: multiply diagonals going down! Step 4: Bottom minus top! 94 - (-8) 94 +8 = 102

55. Example 1 Evaluating the Determinant of a 3x3 Matrix -135 2-46 01-1 Step 1: Rewrite first two columns of the matrix. Step 2: multiply diagonals going up! Step 3: multiply diagonals going down! Step 4: Bottom minus top!

56. Now You Are Asking … How can I do this on the calculator?

57. Step 1: Go to Matrix (above the x-1 key) Step 2: Arrow to the right to EDIT to allow for entering the matrix. Step 3: Type in the dimensions (size) of your matrix and enter the values (press ENTER). Step 4: Go to Matrix again. Notice that your matrix (3x3) now appears showing it in residence.

58. Step 5: Arrow to the right to MATH. Choose #1: det ( Step 6: The function det( will appear on the home screen waiting for a parameter (in this case, the name of the determinant to evaluate). Step 7: Go to Matrix again. Choose [A] or whichever location holds your matrix. Step 8: The name of the determinant's location will appear as the parameter. Hit ENTER to see the evaluation.

59. Example 2 Using a Calculator -135 2-46 01-1 Step 1: Enter matrix A into your calculator. Step 2: Use the matrix submenus to evaluate the determinant of A. 1-12 0 4 2 3 -6 10

60. Identity Matrices  An identity matrix is a square matrix that has 1’s along the main diagonal and 0’s everywhere else.  When you multiply a matrix by the identity matrix, you get the original matrix.

61. Inverse Matrices  When you multiply a matrix and its inverse, you get the identity matrix. If A and B are inverse matrices, then AB=BA=I. Matrix A Inverse of Matrix A Identity X

62. Example 4 Solving a Matrix Equation 002 1 3 -2 1 -2 1 2 0 1 0 1 4 1 0 0 X = 0 -6 19 8 -2 Use the equation X = A -1 C

63. 4-7 Inverse Matrices and Systems What you’ll learn … To solve systems of equations using inverse matrices. 1.04 Operate with matrices to model and solve problems. 2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

64. You can represent a system of equations with a matrix equation. System of Equation x + 2y = 5 3x +5y = 14 Matrix Equation 1 2 3 5 xyxy = 5 14

65. Example 1 and 2 Writing and Solving a System as a Matrix. System of Equation 5a + 3b = 7 3a + 2b = 5 = System of Equation x + 3y = 22 3x + 2y = 10 =

66. Example 3 Solving a System of Three Equations. System of Equation 2x + y +3z = 1 5x + y – 2z = 8 x – y -9z = 5 = System of Equation x – y + z = 0 x – 2y – z = 5 2x - y + 2z = 8 =

67. Example 5 Unique Solutions. System of Equation x + y = 3 x – y = 7 System of Equation 3x + 5y = 1 2x –y = -8 a b c d = ad - bc det Remember…..

68. In Chapter 4, You Should Have… Moved from using matrices in organizing data to manipulating matrices through data. Learned to represent real-world relationships by writing matrices and using operations such as addition and multiplication to develop new matrices.