Matrices. Introduction to Matrices Class Work How many rows and columns does each matrix have? 1. A = ( ) 2.

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1 Matrices Introduction to Matrices How many rows and columns does each matrix have? 1. A = ( ) 2. B = ( 1 0) 0 2. C = ( 5 ) D = ( ) E 5x1 6. F 2x4 Identify the given element using the matrices above.. a 2,1 8. b 1,1 9. c 2,1 10. d 4,2 How many rows and columns does each matrix have? A = ( ) 12. B = ( 5) C = ( 5 9 ) D = ( ) E 2x6 16. F x Identify the given element using the matrices above. 1. a 2, 18. b 2,1 19. c,4 20. d 1,2 Pre-Calc Matrices ~1~ NJCTL.org

2 Adding, Subtracting, and Scalar Multiplication of Matrices A = ( ) B = ( ) C = ( ) D = ( 4) E = ( ) Using the matrices above, perform the given operation or state Not Possible. 21. A 22. 2B 2. A+B 24. C+D 25. D+E 26. C+A 2. A C 28. B A 29. D C 0. 2A + B C A = ( 9 5 ) B = ( 2 9 0) C = ( 2 ) D = ( 1 ) E = ( ) Using the matrices above, perform the given operation or state Not Possible. 1. D 2. 2C. A+D 4. C+B 5. D+E 6. C+A. A C 8. B A 9. E C 40. 2B + A 4C Matrix Multiplication Determine if the indicated multiplication is possible. If it is, give the dimensions of the product. 41. A 2X B X4 42. A 4X5 B 4X5 4. A 1X6 B 6X1 44. A 2X5 B 5X9 45. A X B X4 Perform the following multiplication, or write Not Possible. 46. ( ) ( ) Pre-Calc Matrices ~2~ NJCTL.org

3 4. ( ) (2 2 0 ) 48. ( ) ( ) ( 0 2) ( ) ( 2 2 1) ( ) Determine if the indicated multiplication is possible. If it is, give the dimensions of the product. 51. A 2X4 B X4 52. A 4X5 B 5X4 5. A 1X5 B 5X 54. A 2X5 B X2 55. A 4X B X1 Perform the following multiplication, or write Not Possible. 56. ( ) ( ) 5. ( ) ( ) (1 2 ) ( 5) ( 5) (1 2 ) ( 1 4 4) ( 0 1) Finding Determinants Find the following determinants Pre-Calc Matrices ~~ NJCTL.org

4 Find the following determinants Finding Inverse Matrices Find the inverse of the given matrix. If no inverse exist, explain why. 81. ( ) 82. ( ) 8. ( 1 2 ) 84. ( ) Pre-Calc Matrices ~4~ NJCTL.org

5 85. ( 4 2 ) 86. ( ) 8. ( ) ( 5 4 2) ( 2 2) ( 5 4 2) Find the inverse of the given matrix. If no inverse exist, explain why. 91. ( ) 92. ( ) 9. ( 5 4 ) 94. ( ) 95. ( ) 96. ( 5 4 ) ( 4) ( ) ( 1 4 ) ( 0 1 4) Solving Systems of Equations with Matrices Solve the following systems using matrices x + y = 1 x + y = x 2y = 8x 4y = 14 Pre-Calc Matrices ~5~ NJCTL.org

6 10. 5x + 6y = 2 4x y = x + 5y = 4 2x y = x + y 4z = 11 x y + 2z = x + y + z = x + y = y + z = 2 x + z = x + y + 2z = 10 x + y + z = 1 4x + y + z = 10 Solve the following systems using matrices x + y = 10 x y = x 2y = 8 x 4y = x + y = 6 4x 2y = x + 6y = 2 x y = x y = 9 1x + 2y + 4z = 6 x + y + 2z = x + y = y + z = 2 x + z = x + 2y + 2z = 10 x + y + z = 1 4x + 8y + 8z = 10 Circuits: Definitions and Properties 115. Draw a network that reflects the information in the table Name any loops. 11. Name any parallel edges Is any vertex isolated? If so which? 119. Is this a simple graph? What needs to be done to make it one? 120. What is the degree of each vertex? What is the degree of the network? 121. Create an adjacency matrix for this network. Edge Endpoints e 1 {v 1,v } e 2 {v 2} e {v 4,v 2} e 4 {v,v 4} e 5 {v 1,v } e 6 {v 1,v 4} e {v,v 4} Pre-Calc Matrices ~6~ NJCTL.org

7 122. At holiday cookie exchange everyone gives everyone else half dozen cookies. If 20 people showed up how many cookies were given? 12. At a business meeting with 111 people in attendance is it possible for everyone to shake hands exactly 11 times? 124. Use the following adjacency matrix to create a directed graph. (Rows are the starts) v 1 v 2 v v 4 v 5 v 1 v 2 v v 4 v 5 [ ] 125. Draw a network that reflects that connects v 1, v 2, v, v 4, and v 5 in the table Name any loops. 12. Name any parallel edges Is any vertex isolated? If so, which? 129. Is this a simple graph? What needs to be done to make it one? 10. What is the degree of each vertex? What is the degree of the network? 11. Create an adjacency matrix for this network. 12. At holiday cookie exchange everyone gives everyone else 10 cookies. If 15 people showed up how many cookies were given? 1. At a business meeting with 111 people in attendance is it possible for everyone to shake hands exactly 10 times? 14. Use the following adjacency matrix to create a directed graph. (Rows are the starts) v 1 v 2 v v 4 v 5 v 1 v 2 v v 4 v 5 [ ] Edge Endpoints e 1 {v 2,v } e 2 {v 1} e {v,v 2} e 4 {v,v 4} e 5 {v 2,v } e 6 {v 2,v 4} e {v,v 4} Euler 15. Name a walk from A to C. 16. If edge e was removed find a walk from B to E. 1. Is this graph traversable? 18. Show Euler s Formula holds for this graph. 19. Is the graph a connected graph? 140. Which edges could be removed for it still to be connected? 141. What edges need to be added for this to be an Euler circuit? Pre-Calc Matrices ~~ NJCTL.org

8 142. Name a walk from B to D. 14. If edge f was removed find a walk from A to E Is this graph traversable? 145. Show Euler s Formula holds for this graph Is the graph a connected graph? 14. Which edges could be removed for it still to be connected? 148. What edges need to be added for this to be an Euler circuit? 149. Show that Euler s Formula holds for this graph. Matrix Powers and Walks Given the directed adjacency matrix A, answer the following How many walks of length 2 are there from a 2 to a 4? 151. How many walks of length 2 are there from a 1 to a 4? 152. How many walks of length are there from a 2 to a 4? 15. How many walks of length are there from a 1 to a 4? 154. How many walks of length 4 are there from a 2 to a 4? Given the directed adjacency matrix B, answer the following How many walks of length 2 are there from b 1 to b? 156. How many walks of length 2 are there from b 2 to b? 15. How many walks of length are there from b 1 to b? 158. How many walks of length are there from b 2 to b? 159. How many walks of length 5 are there from b 1 to b? Given the directed adjacency matrix A, answer the following How many walks of length 2 are there from a 2 to a 4? 161. How many walks of length 2 are there from a 1 to a 4? 162. How many walks of length are there from a 2 to a 4? 16. How many walks of length are there from a 1 to a 4? 164. How many walks of length 4 are there from a 2 to a 4? Given the directed adjacency matrix B, answer the following How many walks of length 2 are there from b 1 to b? 166. How many walks of length 2 are there from b 2 to b? 16. How many walks of length are there from b 1 to b? 168. How many walks of length are there from b 2 to b? 169. How many walks of length 6 are there from b 1 to b? Pre-Calc Matrices ~8~ NJCTL.org

9 Markov Chains 10. John, Harold and George are learning to throw a Frisbee at the park. When John throws it he has 50% chance of getting it to George, 25% to Harold, and a 25% it comes back to him. Harold reaches George 40%, John 0%, and himself 0%. George reaches John 0% and Harold 0%. a. Create a matrix to represent this situation. b. Create a vertex-edge graph that models this situation. Label. c. Multiplying the matrix in part (a) with itself will give the percentage for 2 throws. What is the probability that the Frisbee starts with John and ends with George in 2 throws? d. In ten throws, who will have the Frisbee? Does it matter where it started? Explain. 11. A variety of corn can have either grow either one ear per stalk or two. It is known that the kernels from a one eared stalk will grow one eared stalks 65% of the time. The kernels from a two eared stalk will produce two eared stalks 5% of the time. a. Create a matrix to represent this situation. b. Create a vertex-edge graph that models this situation. Label. c. What is the probability that the two eared stalk will have lead to a one eared stalk in generations? d. In ten generations, what are the chances that an unknown kernel will grow a two eared stalk? Does it matter where it started? Explain. Unit Review Multiple Choice 1. Given the matrices at right, what are the dimensions of A? a. x b. 2x c. x2 d. 2x2 2. What operations can be done with matrices A and B? I. Multiplication II. Addition III. Subtraction IV. Scalar Multiplication a. I only b. II and III c. I and IV d. all of the above. What element is 4(A 1,2) a. b. 4 c. 6 d. 8 Pre-Calc Matrices ~9~ NJCTL.org

10 Using the given matrices, perform the indicated operation and answer the question. 4. In A+E, what is the element in the 1,2 position? a. 5 b. c. -9 d. not possible 5. In D B, what is the element in 2, position? a. -4 b. -1 c. 1 d In A*E, what is the element in 1,1 position? a. 6 b. 10 c. 24 d. not possible. C = a. -5 b. - c. d det F = a. 4 b. 8 c. 12 d. not possible 9. Matrix G is 2x2 but does not have an inverse, which of the following is G? a. ( ) b. ( ) c. ( ) d. ( ) Given ( 0 1 4), find x, y, and z a. (2, 4, ) b. (6, -5, ) c. (-4, 1, ) d. cannot be determined Pre-Calc Matrices ~10~ NJCTL.org

11 11. Which is a walk from A to C? a. A h D a A f E e B g C b. A f E e B b C c. A h D c C d. A h D a A f E e B b C 12. What is the degree of B? a. 2 b. c. 4 d At this year s Knowledge Slam there were 8 teams in attendance. During the opening rounds every team went against every team to determine who would go on to the semifinals. How many total meetings were there before the semifinals? a. 4 b. c. 28 d How many ways are there to go from b 1 to b 2 of length 6? a. 0 b c d Extended Response 1. Alice, Bob, and Chris go get ice cream. Alice gets flavors, 2 hot toppings and 1 cold topping. Bob gets 2 flavors, 1 hot and 1 cold topping. Chris gets flavors, 1 hot topping and cold. a. Create a matrix to represent what they purchased, let each person have their own row. b. They make this visit on a regular basis, getting the same number of flavors and toppings but they like to order different flavors. What is the most of each that they could order after 4 visits? Answer can be left in matrix form. c. What operation was used in part b? Would this operation still be possible if one of them missed a visit? 2. John, Harold and George are learning to throw a Frisbee at the park. When John throws it he has 40% chance of getting it to George, 5% to Harold, and a 25% it comes back to him. Harold reaches George 50%, John 0%, and himself 20%. George reaches John 60% and Harold 40%. a. Create a matrix to represent this situation. b. Create a vertex-edge graph that models this situation. Label. Pre-Calc Matrices ~11~ NJCTL.org

12 c. Multiplying the matrix in part (a) with itself will give the percentage for 2 throws. What is the probability that the Frisbee starts with John and ends with George in 2 throws?. Create a vertex edge graph that meets the following conditions. 5 vertices labeled A thru E 11 edges labeled a to k directed a loop at B 2 ways from C to D and 1 from D to C A is isolated a. create an adjacency matrix for your graph b. Does your graph have any parallel edges? If so name them, if not explain why. c. Name a circuit starting at C, if one exists rows, 2 columns 2. 1 row, columns. rows, 2 columns 4. 4 rows, columns 5. 5 rows, 1 column 6. 2 rows, 4 columns rows, columns 12. rows, 1 column 1. rows, 4 columns 14. rows, columns rows, 6 columns 16. rows, columns ( ) ( ) Answers 6 2. ( 2 10) not possible 25. not possible ( 0 12) ( 0 4) ( 2 2 ) not possible ( ) ( 9 ) ( ) not possible ( 4 12 ) not possible Pre-Calc Matrices ~12~ NJCTL.org

13 ( ) ( 8 14) ( 14 ) not possible ( ) yes, 2x4 42. no 4. yes, 1x1 44. yes, 2x9 45. yes, x ( 2 6 ) ( 1 8 ) 48. not possible ( ) ( 5 16 ) no 52. yes, 4x4 5. yes, 1x 54. no 55. yes, 4x ( 4 9 ) 5. ( ) 58. (2) ( ) ( ) ( ) 82. ( ) 8. ( ) 84. not possible, det=0 85. ( 2 4 ) (.5 ) 8. not possible, not square ) ( ( ) ( (.5 1 ) ( 1 2 ) 9. ( ) ) Pre-Calc Matrices ~1~ NJCTL.org

14 94. not possible, det= ( ) ( 4 ) 9. not possible, not square ( ) 99. ( ) ( ) (2, -1) 102. (2, ½) 10. (2, -2) 104. (1, 0) 105. (2, 1,-1) 106. (8, -1, ) 10. (1,, ) 108. (4, 2) 109. (-1,) 110. (0, 2) 111. (.5, -1.5) 112. (4, -1, 1) 11. (5, -1, -1) 114. no solution 115. Answers will vary 116. E2 11. E1 e5; e4 e 118. No 119. Need to eliminate e2, e, or e5, and e4 or e 120. V1: ; V2: 4; V: 4; V4: 4; network: [ ] cookies 12. No, not possible to have an odd number of odd vertices 124. Answers will vary 125. Answers will vary 126. E2 12. E1 e e5; e4 e 128. Yes, V No loops, no parallel 10. V1:2; V2:4; V:5; V4:; V5:0; network: [ ] cookies 1. Yes 14. Answers will vary 15. A e E c C 16. B d D f E 1. Yes 18. V=5; E=8 F=5: 5-8+5=2 19. Yes 140. Answers will vary 141. Already is one 142. B e E f A a D 14. A h D d C c B e E 144. No more than 2 odd vertices 145. V=5; E=8; F=5: f-8+5= Yes 14. b, c, or do 148. all vertices need to be even 149. v=6; E=6; F=2: 6-6+2= Pre-Calc Matrices ~14~ NJCTL.org

15 , , A. [ ] B J M..5.. G.4. C. 22.5% D. J A. [ ] b E EE.5.5 c. 9% d. 58 Pre-Calc Matrices ~15~ NJCTL.org