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MatricesIntroduction to MatricesClass WorkHow many rows and columns does each matrix have?

1. A=( 2 4−1 5)

2. B=(3 1 0 )

3. C=( 0 −23 5

−1 4 )4. D=(

2 4 51 4 80 2 31 6 3)

5. E5 x16. F2x 4

Identify the given element using the matrices above.7. a2,18. b1,19. c2,110. d4,2

HomeworkHow many rows and columns does each matrix have?

11. A=(2 4 51 7 90 8 31 5 20 4 6

)12. B=(356)13. C=( 0 −2

3 5−1 4

−4 69 71 0)

14. D=(4 −1 52 9 21 0 −7)

15. E2 x616. F3x 7

Pre-Calc Matrices ~1~ NJCTL.org

Identify the given element using the matrices above.17. a2,318. b2,119. c3,420. d1,2

Adding, Subtracting, and Scalar Multiplication of MatricesClass Work

A=( 2 3 50 −2 4

−1 2 −6)B=( 1 3 −82 −5 6

−3 4 0 )C=(−1 2 40 −5 86 2 −3)D=(2 1

3 40 9)E=(2 3 5

1 7 9)Using the matrices above, perform the given operation or state “Not Possible”.21. 3A22. 2B23. A+B24. C+D25. D+E26. C+A27. A – C28. B – A29. D – C30. 2A + B – 3C

Homework

A=(−1 2 59 −5 −71 2 −5)B=(1 3 6

2 9 03 4 5)C=(4 −4 9

2 3 71 0 8)D=(−9 0

1 3−7 4)E=(2 3 4

1 5 6)Using the matrices above, perform the given operation or state “Not Possible”.31. 3D32. 2C33. A+D34. C+B35. D+E36. C+A37. A – C38. B – A39. E – C40. 2B + 3A – 4C

Matrix MultiplicationClass WorkDetermine if the indicated multiplication is possible. If it is, give the dimensions of the product.41. A2 X3×B3 X 4


42. A4X 5×B4 X 543. A1 X6×B6 X 144. A2 X5×B5 X 945. A7 X 3×B3 X 4

Perform the following multiplication, or write “Not Possible”.

46. (2 40 1 )(3 5

2 6)47. (1 −1

2 3 )(2 −43 0 )

48. (2 3 10 1 1)(4 −1 2

1 0 2)49. ( 1 3

0 2−1 4)(1 3 −2

5 6 −1)

50. ( 1 3 0−2 2 14 1 3)(

2 3 −72 −1 03 3 2 )

HomeworkDetermine if the indicated multiplication is possible. If it is, give the dimensions of the product.51. A2 X 4×B3X 452. A4X 5×B5X 453. A1 X5×B5 X354. A2 X5×B3 X255. A4X 3×B3X 1

Perform the following multiplication, or write “Not Possible”.

56. (1 32 −1)( 0 −2

−4 5 )57. (1 2

4 0)(−9 22 3)

58. (1 2 3 )(456)59. (456) (1 2 3 )

60. ( 2 −2 3−1 −4 40 2 0)(

4 −2 43 0 −1

−2 3 0 )


Finding DeterminantsClass WorkFind the following determinants.

61. |4 32 5|

62. |−1 43 5|

63. |6 −83 4 |

64. | 2 −5−3 8 |

65. |4 72 3|

66. | 4 6−2 0|

67. | 1 0 23 4 1

−6 2 3|68. |2 1 0

3 4 12 0 1|

69. |1 −1 20 2 32 0 4|

70. | 2 2 −11 5 2

−6 4 3 |HomeworkFind the following determinants.

71. |−2 32 4|

72. | 1 4−2 3|

73. |3 42 0|

74. |5 102 4 |

75. |−1 5−3 2|


76. | 2 −6−3 4 |

77. |1 0 14 3 25 1 2|

78. |1 1 12 3 40 5 −2|

79. | 2 −1 3−3 0 45 1 −2|

80. | 2 3 −10 −2 0

−1 4 2 |Finding Inverse MatricesClass WorkFind the inverse of the given matrix. If no inverse exist, explain why.

81. (2 34 1)

82. (4 53 4)

83. (−1 23 3)

84. ( 4 −2−6 3 )

85. (4 33 2)

86. (6 82 3)

87. (1 4 55 3 0)

88. (2 0 15 4 21 −1 2)

89. (1 1 13 2 24 2 0)


90. (2 3 15 4 20 0 2)

HomeworkFind the inverse of the given matrix. If no inverse exist, explain why.

91. (2 41 3)

92. (4 52 3)

93. (−5 3−4 3)

94. (−1 22 −4)

95. (3 54 6)

96. (−3 54 −7)

97. (1 23 44 5)

98. ( 1 2 −4−1 1 52 7 −3)

99. (1 3 31 4 31 3 4 )

100. (1 2 30 1 45 6 0)

Solving Systems of Equations with MatricesClass WorkSolve the following systems using matrices.101. 2 x+3 y=1

3 x+ y=5102. 4 x−2 y=7

8 x−4 y=14103. 5 x+6 y=−2

4 x− y=10104. 4 x+5 y=4


2 x− y=2105. 2 x+3 y−4 z=11

3 x− y+2 z=3x+ y+z=2

106. x+ y=7y+z=2x+z=11

107. x+ y+2 z=10x+3 y+ z=134 x+ y+z=10

HomeworkSolve the following systems using matrices.108. x+3 y=10

3 x− y=10109. 2 x−2 y=−8

3 x−4 y=−15110. 5 x+3 y=6

4 x−2 y=−4111. 2 x+6 y=−2

x− y=5112. 2 x− y=9

1 x+2 y+4 z=6x+ y+2 z=5

113. 2 x+3 y=7y+3 z=2x+z=6

114. x+2 y+2 z=10x+3 y+ z=134 x+8 y+8 z=10

Circuits: Definitions and PropertiesClass Work115. Draw a network that reflects the information in the table.116. Name any loops.117. Name any parallel edges.118. 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.122. At holiday cookie exchange everyone gives everyone else half dozen cookies. If 20

people showed up how many cookies were given?


Edge Endpointse1 {v1,v3}e2 {v2}e3 {v4,v2}

e4 {v3,v4}e5 {v1,v3}e6 {v1,v4}e7 {v3,v4}

123. 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) v1 v2 v3 v4 v5

v1v2v3v4v5

[1 0 00 1 3

1 11 0

1 1 00 0 01 1 0

2 10 01 0

]Homework125. Draw a network that reflects that connects v1, v2, v3, v4, and v5 in

the table.126. Name any loops.127. Name any parallel edges.128. Is any vertex isolated? If so, which?129. Is this a simple graph? What needs to be done to make it one?130. What is the degree of each vertex? What is the degree of the

network?131. Create an adjacency matrix for this network.132. At holiday cookie exchange everyone gives everyone else 10 cookies. If 15 people

showed up how many cookies were given?133. At a business meeting with 111 people in attendance is it possible for everyone to shake

hands exactly 10 times?134. Use the following adjacency matrix to create a directed graph. (Rows are the starts) v1 v2 v3 v4 v5

v1v2v3v4v5

[0 1 20 0 2

0 10 1

1 2 00 1 21 0 0

0 10 00 1

]EulerClass Work135. Name a walk from A to C. 136. If edge e was removed find a walk from B to E.137. Is this graph traversable?138. Show Euler’s Formula holds for this graph.

139. Is the graph a connected graph?140. Which edges could be removed for it still to be connected?


Edge Endpointse1 {v2,v3}e2 {v1}e3 {v3,v2}

e4 {v3,v4}e5 {v2,v3}e6 {v2,v4}e7 {v3,v4}

141. What edges need to be added for this to be an Euler circuit? Homework142. Name a walk from B to D. 143. If edge f was removed find a walk from A to E. 144. Is this graph traversable?145. Show Euler’s Formula holds for this graph.

146. Is the graph a connected graph?147. 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 WalksClass WorkGiven the directed adjacency matrix A, answer the following.150. How many walks of length 2 are there from a2 to a4?151. How many walks of length 2 are there from a1 to a4?152. How many walks of length 3 are there from a2 to a4?153. How many walks of length 3 are there from a1 to a4?154. How many walks of length 4 are there from a2 to a4?Given the directed adjacency matrix B, answer the following.155. How many walks of length 2 are there from b1 to b3?156. How many walks of length 2 are there from b2 to b3?157. How many walks of length 3 are there from b1 to b3?158. How many walks of length 3 are there from b2 to b3?159. How many walks of length 5 are there from b1 to b3?

HomeworkGiven the directed adjacency matrix A, answer the following.160. How many walks of length 2 are there from a2 to a4?161. How many walks of length 2 are there from a1 to a4?162. How many walks of length 3 are there from a2 to a4?163. How many walks of length 3 are there from a1 to a4?164. How many walks of length 4 are there from a2 to a4?Given the directed adjacency matrix B, answer the following.165. How many walks of length 2 are there from b1 to b3?166. How many walks of length 2 are there from b2 to b3?167. How many walks of length 3 are there from b1 to b3?168. How many walks of length 3 are there from b2 to b3?169. How many walks of length 6 are there from b1 to b3?


Markov ChainsClass Work170. 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 30%, and himself 30%. George reaches John 70% and Harold 30%.

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.

Homework171. 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 75% 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

3 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 ReviewMultiple Choice

1. Given the matrices at right, what are the dimensions of A?a. 3x3b. 2x3c. 3x2d. 2x2

2. What operations can be done with matrices A and B?I. Multiplication II. Addition III. Subtraction IV. Scalar Multiplication

a. I onlyb. II and IIIc. I and IVd. all of the above

3. What element is 4(A1,2)a. 3b. 4c. 6


d. 8

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. 5b. 7c. -9d. not possible

5. In D – B, what is the element in 2,3 position?a. -4b. -1c. 1d. 4

6. In A*E, what is the element in 1,1 position?a. 6b. 10c. 24d. not possible

7. |C|=a. -5b. -3c. 3d. 5

8. det F =a. 4b. 8c. 12d. not possible

9. Matrix G is 2x2 but does not have an inverse, which of the following is G?

a. (3 24 0)

b. (1 61 5)

c. ( 4 210 5)

d. (6 −38 4 )

10. Given (1 2 20 1 30 0 1|

243 ) , find x , y ,∧z .


a. (2, 4, 3)b. (6, -5, 3)c. (-4, 1, 3) d. cannot be determined

11. Which is a walk from A to C?a. A→h→D→a→A→f→E→e→B→g→Cb. A→f →E→e→B→b→Cc. A→h→D→c→Cd. A→h→D→a→A→f→E→e→B→b→C

12. What is the degree of B?a. 2b. 3c. 4d. 5

13. 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. 4b. 7c. 28d. 56

14. How many ways are there to go from b1 to b2 of length 6?a. 30b. 15625c. 676587d. 761684

Extended Response1. Alice, Bob, and Chris go get ice cream. Alice gets 3 flavors, 2 hot toppings and 1 cold

topping. Bob gets 2 flavors, 1 hot and 1 cold topping. Chris gets 3 flavors, 1 hot topping and 3 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, 35% to Harold, and a 25% it comes back to


him. Harold reaches George 50%, John 30%, 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.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?

3. 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 graphb. 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.

Answers1. 2 rows, 2 columns2. 1 row, 3 columns3. 3 rows, 2 columns4. 4 rows, 3 columns5. 5 rows, 1 column6. 2 rows, 4 columns7. -18. 39. 310. 611. 5 rows, 3 columns12. 3rows, 1 column13. 3 rows, 4 columns14. 3 rows, 3 columns15. 2 rows, 6 columns16. 3 rows,7 columns17. 918. 519. 020. -1

21. ( 3 9 150 −6 12

−3 6 −18)

22. ( 2 6 −164 −10 12

−6 8 0 )23. ( 3 6 −3

2 −7 10−4 6 −6)

24. not possible25. not possible

26. (1 5 90 −7 125 4 −9)

27. ( 3 1 10 3 −4

−7 0 −3 )28. (−1 0 −13

2 −3 2−2 2 6 )

29. not possible

30. ( 8 3 −102 −24 −10

−13 2 −3 )Pre-Calc Matrices ~13~ NJCTL.org

31. (−27 03 9

−21 12)32. (8 −8 18

4 6 142 0 16)

33. not possible

34. (5 −1 154 12 74 4 13)

35. not possible

36. ( 3 −2 1411 −2 02 2 3 )

37. (−5 6 −47 −8 −140 2 −13)

38. ( 2 1 1−7 14 72 2 10)

39. not possible

40. (−17 28 −923 −9 −495 14 −37)

41. yes, 2x442. no43. yes, 1x144. yes, 2x945. yes, 7x4

46. (14 342 6 )

47. (−1 −413 −8)

48. not possible

49. (16 21 −510 12 −219 21 −2)

50. ( 8 0 −73 −5 1619 20 −22)

51. no

52. yes, 4x453. yes, 1x354. no55. yes, 4x1

56. (−12 134 −9)

57. ( −5 8−36 8)

58. (32)

59. (4 8 125 10 156 12 18)

60. ( −4 5 10−24 14 86 0 −2)

61. 1462. -1763. 4864. 165. -266. 1267. 7068. 769. -670. -5071. -1472. 1173. -874. 075. 1376. -1077. -778. -1279. -3180. -6

81. (−.1 .3.4 −.2)

82. ( 4 −5−3 4 )


83. (−13

29

13

19)

84. not possible, det=0

85. (−2 33 −4 )

86. ( 1.5 −2−.5 3 )

87. not possible, not square

88. (1011

−111

−411

−811

311

111

−911

211

811

)89. (−2 1 0

4 −2 .5−1 1 −.5)

90. (−47

37

−17

57

−27

−114

0 0 12

)91. ( 1.5 −2

−.5 1 )92. (1.5 −2.5

−1 2 )93. (−1 1

−43

53 )

94. not possible, det=0

95. (−3 2.52 −1.5)

96. (−7 −5−4 −3)

97. not possible, not square

98. (−196

−116

76

712

512

−112

−34

−14

14

)99. ( 7 −3 −3

−1 1 0−1 0 1 )

100. (−24 18 520 −15 −4−5 4 1 )

101. (2, -1)102. (2, ½)103. (2, -2)104. (1, 0)105. (2, 1 ,-1)106. (8, -1, 3)107. (1, 3, 3)108. (4, 2)109. (-1,3)110. (0, 2)111. (3.5, -1.5)112. (4, -1, 1)113. (5, -1, -1)114. no solution

115. Answers will vary116. E2

117. E1 ||e5; e4||e7

118. No119. Need to eliminate e2, e, or e5, and e4 or e7

120. V1: 3; V2: 4; V3: 4; V4: 4; network: 14

121. [0021010120021120]

122. 1140cookies123. No, not possible to have an odd number of odd vertices

124. Answers will vary


125. Answers will vary126. E2

127. E1 || e3||e5; e4||e7

128. Yes, V5

129. No loops, no parallel130. V1:2; V2:4; V3:5; V4:3; V5:0; network: 14

131. [1000000310030200120000000

]132. 1050cookies133. Yes134. Answers will vary135. A eEcC136. BdDfE137. Yes138. V=5; E=8 F=5: 5-8+5=2139. Yes140. Answers will vary141. Already is one142. BeEfAaD143. AhDdCcBeE144. No more than 2 odd vertices145. V=5; E=8; F=5: f-8+5=2146. Yes147. b, c, or do148. all vertices need to be even149. v=6; E=6; F=2: 6-6+2=2150. 16151. 11152. 70153. 96154. 796155. 43156. 26157. 1789


158. 1542159. 387,049160. 9161. 4162. 18163. 82164. 333165. 36166. 38167. 435168. 506169. 764,268

170. A. [ .25.25 .5.3 .3 .4.7 .30 ]

B. .25 .25J M .3

.5 .3 .7 G .4 .3

C. 22.5%’D. J 40

171. A. [ .65.35.25.75 ]b. .25 .65 E EE .75

.35c. 39%d. 58


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