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Chapter 4

Chapter 4. Matrices. 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.

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Chapter 4

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  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. 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. 4-1 Organizing Data

  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. Columns Rows

  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 a21 = 1, since the element in the 2nd row and 1st column is 1. The element a13 = 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. a33 b. a11 c. a21 d. a12 • 17 24 3 • 10.4 12 15 • 9 30 15

  8. Write a matrix W to represent the information. Which element represents Maloney’s score on the vault? Example 4a Real World Example

  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 m32 represent?

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

  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. 1 -2 0 3 9 -3 3 -5 7 -9 6 12 -12 24 -3 1 -3 5 2 -4 -1 10 -1 5 Example Adding Matrices

  14. The additive identity matrix for the set of all m x n matrices is the zero matrix 0, or Omxn ,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 • -4 • -1 0 -2 4 1 0 A = -A =

  15. 1 2 0 0 5 -7 0 0 2 8 -2 -8 -3 0 3 0 Example 2 Using Identity and Inverse Matrices

  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)=0 Additive Inverse Property

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

  18. 6 -9 7 -4 3 0 -2 1 8 6 5 10 Example 3 Subtracting Matrices -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. -1 0 2 5 10 7 -4 4 X + =

  20. Solve X - = Solve X + = Example 4 Solving A Matrix Equation • 1 • 3 2 0 1 8 9 -1 0 2 5 • 7 • -4 4

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

  22. 2x-5 4 25 4 3 3y+12 3 y+18 Example 6 Finding Unknown Matrix Elements X = ____ Y = ____ x+8 -5 38 -5 3 -y 3 4y-10 X = ____ Y = ____

  23. 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. 4-3 Matrix Multiplication

  24. 5 • 2 8 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. 9 15 6 24 3 =

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

  26. Example 2 Using Scalar Products 3 0 6 -1 8 2 2 3 -7 1 4 5 A= B= Find 5B- 4A Find 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 10 0 4 2 3 4 -2 1 = 4x + 2

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

  30. How much money did the cafeteria collect selling lunch 1? Selling Lunch 2? Selling Lunch 3? 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. 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. Investigation: Using Matrices

  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. Dimensions: 3 x 2 2 x 3 **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. They must match. The dimensions of your answer.

  33. **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 Dimensions: 2 x 3 2 x 2 *They don’t match so can’t be multiplied together.*

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

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

  36. 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. Example 4a Multiplying Matrices -1 0 3 -4 -3 3 5 0

  37. 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. Example 4b Multiplying Matrices -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 9 30 20 Tapes LPs CDs $8 $6 $13

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

  40. Example 6a Determining When a Product Matrix Exists • 0 • 2 -5 • 3 • -1 8 • 4 0 Use matrices G = and H = . Determine whether products GH and HG are defined (exist) or undefined (do not exist).

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

  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. 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. 4-5 2x2 Matrices, Determinants, and Inverses

  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 • 3 • 1 2 • -3 • -1 2 A = B = 3 -1 7 -1 .1 .1 -.7 .3 M = N =

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

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

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

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