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Chapter 4

Chapter 4. Matrices. 4.1 Intro to Matrices. Matrix: a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets Element: a value in a matrix Dimensions: number of rows x number of columns Read “m by n”.

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Chapter 4

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  1. Chapter 4 Matrices

  2. 4.1 Intro to Matrices • Matrix: a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets • Element: a value in a matrix • Dimensions: number of rows x number of columns • Read “m by n”

  3. State the dimensions of matrix G if G = • State the dimensions of matrix A if A =

  4. Types of Matrices • Row matrix: • A matrix with only one row ex: • Column matrix: • A matrix with only one column ex: • Square matrix: • A matrix with the same number of rows as columns • ex: • Zero matrix: • All elements are zero

  5. Solve an equation involving Matrices 1. =

  6. 2.

  7. 3. • Write a matrix for the prices of movie tickets for adults, children, and seniors. • What are the dimensions of the matrix?

  8. 4.2 Operations with matrices

  9. Matrices can be added and subtracted if, and only if, they have the same dimensions. • ex: + = • ex: =

  10. 1. Find A + B if A = and B =

  11. 2. Find A + B if A = and B =

  12. 3. Find B – A if A = and B =

  13. Scalar Multiplication • Scalar: a constant that you can multiply a matrix by • ex: =

  14. 4. Find 3A, if A =

  15. 5. If A = and B = , find 5A – 2B

  16. 4.3 Multiplying matrices

  17. You can multiply matrices if and only if: the number of columns in the first matrix is the same as the number of rows in the second matrix • Ex: A5 x3 and B3x4 = AB • If the matrices cannot be multiplied = product matrix is not defined 5 x 4

  18. Multiplying Matrices x = Step 1: x Step 2 : x Step 3 : x Step 4 : x

  19. Find RS if 1. R = and S =

  20. At a swimming meet 6 points are awarded for 1st place, 4 points for 2nd place, and 3 points for 3rd place. 2. The chart shows how many swimmers placed in each position through the meet for the four participating schools. Write a set of matrices to model the points earned. Which team won the meet?

  21. Commutative Property – Does it work for matrices? 3. Find each product if P = and S = a. PS b. SP

  22. Distributive Property – Does it work for matrices? 4. Find each product if A = B = and C = a. A (B + C) b. AB + AC

  23. 4.5 Determinants

  24. Determinant: A number associated with a square matrix • Second-Order Determinant • A value found by calculating the difference of the products of the two diagonals in a 2x2 matrix • = ad – bc

  25. Find the value of the determinant 1. 2.

  26. 3. 4.

  27. Third-Order Determinant • Determinant of a 3x3 matrix • Method 1: Expansion by Minors • = a - b + c • Method 2: Diagonals

  28. Find the determinant using expansion by minors 3.

  29. 4.6 Cramer’s rule

  30. Use the determinants to solve systems of equations • Ex: ax + by = e cx + dy = f x = and y = Write the answer as (x, y) x = and y =

  31. Solve the system of equations using Cramer’s rule 1. 5x + 4y = 28 3x – 2y = 8

  32. 2. 2x – 3y = 12 -6x + y = -20

  33. In voting for the colors of a new high school, blue & gold received 440 votes from 10th and 11th graders while red & black received 210 votes from the same grades. In the 10th grade, blue & gold received 72% of the total and Red & black received 28%. In the 11th grade, Blue & gold received 64% of the total and Red & Black received 36%. • Write a system of equations that represents the total number of votes for each pair of colors. • Find the total number of votes cast in 10th grade and in 11th grade.

  34. 4.7 Identity and Inverse Matrices

  35. Identity Matrix: • A square matrix that, when multiplied by another matrix equals the same matrix • Ex: or

  36. Inverse Matrices: • When the product of two matrices with the same dimensions is the identity matrix

  37. Determine whether each pair of matrices are inverses of each other. 1. X = and Y =

  38. 2. C = and D =

  39. To find the inverse of a matrix • Find the determinant to see if it has an inverse • If the determinant is zero, it cannot have an inverse • If the inverse exists it =

  40. Find the inverse for the given matrix 3. R =

  41. 4. A =

  42. 4.8 Using Matrices to solve systems of equations

  43. Step 1: Rewrite the system of equations as a matrix equation • Ex: 5x + 7y = 11 3x + 8y = 18 • Step 2: find the inverse matrix • = =

  44. Step 3: Multiply each side of the matrix equation by the inverse matrix • = • Step 4: Write the solution as an ordered pair :

  45. Solve the system using matrices 1. 5x + 3y = 13 4x + 7y = -8

  46. 2. 6a – 9b = -18 8a – 12b = 24

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