Download Matrices

Matrices
Introduction to Matrices
Class Work
How many rows and columns does each matrix have?
2 4
1. A = (
)
−1 5
2. 𝐵 = (3 1 0)
0 −2
3. 𝐶 = ( 3
5)
−1 4
2 4 5
1 4 8
4. 𝐷 = (
)
0 2 3
1 6 3
5. 𝐸5𝑥1
6. 𝐹2𝑥4
Identify the given element using the matrices above.
7. 𝑎2,1
8. 𝑏1,1
9. 𝑐2,1
10. 𝑑4,2
Homework
How many rows and columns does each matrix have?
2 4 5
1 7 9
11. A = 0 8 3
1 5 2
( 0 4 6)
3
12. 𝐵 = (5)
6
0 −2
4 6
13. 𝐶 = ( 3
5 − 9 7)
−1 4
1 0
4 −1 5
14. 𝐷 = (2 9
2)
1 0 −7
15. 𝐸2𝑥6
16. 𝐹3𝑥7
Identify the given element using the matrices above.
17. 𝑎2,3
18. 𝑏2,1
19. 𝑐3,4
20. 𝑑1,2
Pre-Calc Matrices
~1~
NJCTL.org
Adding, Subtracting, and Scalar Multiplication of Matrices
Class Work
1
3 −8
−1 2
4
2 1
2
3
5
2
𝐴 = ( 0 −2 4 ) 𝐵 = ( 2 −5 6 ) 𝐶 = ( 0 −5 8 ) 𝐷 = (3 4) 𝐸 = (
1
−3 4
0
6
2 −3
0 9
−1 2 −6
Using the matrices above, perform the given operation or state “Not Possible”.
21. 3A
22. 2B
23. A+B
24. C+D
25. D+E
26. C+A
27. A – C
28. B – A
29. D – C
30. 2A + B – 3C
Homework
1 3 6
4 −4 9
−9 0
−1 2
5
2
𝐴 = ( 9 −5 −7) 𝐵 = (2 9 0) 𝐶 = (2 3 7) 𝐷 = ( 1 3) 𝐸 = (
1
3 4 5
1 0 8
−7 4
1
2 −5
Using the matrices above, perform the given operation or state “Not Possible”.
31. 3D
32. 2C
33. A+D
34. C+B
35. D+E
36. C+A
37. A – C
38. B – A
39. E – C
40. 2B + 3A – 4C
3 5
)
7 9
3 4
)
5 6
Matrix Multiplication
Class Work
Determine if the indicated multiplication is possible. If it is, give the dimensions of the product.
41. 𝐴2𝑋3 × 𝐵3𝑋4
42. 𝐴4𝑋5 × 𝐵4𝑋5
43. 𝐴1𝑋6 × 𝐵6𝑋1
44. 𝐴2𝑋5 × 𝐵5𝑋9
45. 𝐴7𝑋3 × 𝐵3𝑋4
Perform the following multiplication, or write “Not Possible”.
2 4 3 5
46. (
)(
)
0 1 2 6
Pre-Calc Matrices
~2~
NJCTL.org
1 −1 2 −4
47. (
)(
)
2 3
3 0
2 3 1 4 −1 2
48. (
)(
)
0 1 1 1 0 2
1 3
1 3 −2
49. ( 0 2) (
)
5 6 −1
−1 4
1 3 0 2 3 −7
50. (−2 2 1) (2 −1 0 )
4 1 3 3 3
2
Homework
Determine if the indicated multiplication is possible. If it is, give the dimensions of the product.
51. 𝐴2𝑋4 × 𝐵3𝑋4
52. 𝐴4𝑋5 × 𝐵5𝑋4
53. 𝐴1𝑋5 × 𝐵5𝑋3
54. 𝐴2𝑋5 × 𝐵3𝑋2
55. 𝐴4𝑋3 × 𝐵3𝑋1
Perform the following multiplication, or write “Not Possible”.
1 3
0 −2
56. (
)(
)
2 −1 −4 5
1 2 −9 2
57. (
)(
)
4 0
2 3
4
58. (1 2 3) (5)
6
4
59. (5) (1 2 3)
6
2 −2 3
4 −2 4
60. (−1 −4 4) ( 3
0 −1)
0
2 0 −2 3
0
Finding Determinants
Class Work
Find the following determinants.
4 3
61. |
|
2 5
−1 4
62. |
|
3 5
6 −8
63. |
|
3 4
2 −5
64. |
|
−3 8
4 7
65. |
|
2 3
4 6
66. |
|
−2 0
Pre-Calc Matrices
~3~
NJCTL.org
1 0 2
67. | 3 4 1|
−6 2 3
2 1 0
68. |3 4 1|
2 0 1
1 −1 2
69. |0 2 3|
2 0 4
2 2 −1
70. | 1 5 2 |
−6 4 3
Homework
Find the following determinants.
−2 3
71. |
|
2 4
1 4
72. |
|
−2 3
3 4
73. |
|
2 0
5 10
74. |
|
2 4
−1 5
75. |
|
−3 2
2 −6
76. |
|
−3 4
1 0 1
77. |4 3 2|
5 1 2
1 1 1
78. |2 3 4 |
0 5 −2
2 −1 3
79. |−3 0
4|
5
1 −2
2
3 −1
80. | 0 −2 0 |
−1 4
2
Finding Inverse Matrices
Class Work
Find the inverse of the given matrix. If no inverse exist, explain why.
2 3
81. (
)
4 1
4 5
82. (
)
3 4
−1 2
83. (
)
3 3
4 −2
84. (
)
−6 3
Pre-Calc Matrices
~4~
NJCTL.org
4
85. (
3
6
86. (
2
1
87. (
5
2
88. (5
1
1
89. (3
4
2
90. (5
0
3
)
2
8
)
3
4 5
)
3 0
0 1
4 2)
−1 2
1 1
2 2)
2 0
3 1
4 2)
0 2
Homework
Find the inverse of the given matrix. If no inverse exist, explain why.
2 4
91. (
)
1 3
4 5
92. (
)
2 3
−5 3
93. (
)
−4 3
−1 2
94. (
)
2 −4
3 5
95. (
)
4 6
−3 5
96. (
)
4 −7
1 2
97. (3 4)
4 5
1 2 −4
98. (−1 1 5 )
2 7 −3
1 3 3
99. (1 4 3)
1 3 4
1 2 3
100. (0 1 4)
5 6 0
Solving Systems of Equations with Matrices
Class Work
Solve the following systems using matrices.
101. 2𝑥 + 3𝑦 = 1
3𝑥 + 𝑦 = 5
102. 4𝑥 − 2𝑦 = 7
8𝑥 − 4𝑦 = 14
Pre-Calc Matrices
~5~
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103. 5𝑥 + 6𝑦 = −2
4𝑥 − 𝑦 = 10
104. 4𝑥 + 5𝑦 = 4
2𝑥 − 𝑦 = 2
105. 2𝑥 + 3𝑦 − 4𝑧 = 11
3𝑥 − 𝑦 + 2𝑧 = 3
𝑥+𝑦+𝑧 =2
106. 𝑥 + 𝑦 = 7
𝑦+𝑧 =2
𝑥 + 𝑧 = 11
107. 𝑥 + 𝑦 + 2𝑧 = 10
𝑥 + 3𝑦 + 𝑧 = 13
4𝑥 + 𝑦 + 𝑧 = 10
Homework
Solve the following systems using matrices.
108. 𝑥 + 3𝑦 = 10
3𝑥 − 𝑦 = 10
109. 2𝑥 − 2𝑦 = −8
3𝑥 − 4𝑦 = −15
110. 5𝑥 + 3𝑦 = 6
4𝑥 − 2𝑦 = −4
111. 2𝑥 + 6𝑦 = −2
𝑥−𝑦 =5
112. 2𝑥 − 𝑦 = 9
1𝑥 + 2𝑦 + 4𝑧 = 6
𝑥 + 𝑦 + 2𝑧 = 5
113. 2𝑥 + 3𝑦 = 7
𝑦 + 3𝑧 = 2
𝑥+𝑧 =6
114. 𝑥 + 2𝑦 + 2𝑧 = 10
𝑥 + 3𝑦 + 𝑧 = 13
4𝑥 + 8𝑦 + 8𝑧 = 10
Circuits: Definitions and Properties
Class Work
115. 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.
Pre-Calc Matrices
~6~
Edge
e1
e2
e3
e4
e5
e6
e7
Endpoints
{v1,v3}
{v2}
{v4,v2}
{v3,v4}
{v1,v3}
{v1,v4}
{v3,v4}
NJCTL.org
122. At holiday cookie exchange everyone gives everyone else half dozen cookies. If 20
people showed up how many cookies were given?
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
𝑣1 1 0 0 1 1
𝑣2 0 1 3 1 0
𝑣3 1 1 0 2 1
𝑣4 0 0 0 0 0
𝑣5 [1 1 0 1 0]
Homework
125. Draw a network that reflects that connects v1, v2, v3, v4, and v5 in
Edge Endpoints
the table.
e1
{v2,v3}
e2
{v1}
126. Name any loops.
e3
{v3,v2}
127. Name any parallel edges.
e4
{v3,v4}
128. Is any vertex isolated? If so, which?
e
{v2,v3}
5
129. Is this a simple graph? What needs to be done to make it one?
e6
{v2,v4}
130. What is the degree of each vertex? What is the degree of the
e
{v3,v4}
7
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
𝑣1 0 1 2 0 1
𝑣2 0 0 2 0 1
𝑣3 1 2 0 0 1
𝑣4 0 1 2 0 0
𝑣5 [1 0 0 0 1]
Euler
Class Work
135. 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.
140.
141.
Is the graph a connected graph?
Which edges could be removed for it still to be connected?
What edges need to be added for this to be an Euler circuit?
Pre-Calc Matrices
~7~
NJCTL.org
Homework
142. 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.
147.
148.
149.
Is the graph a connected graph?
Which edges could be removed for it still to be connected?
What edges need to be added for this to be an Euler circuit?
Show that Euler’s Formula holds for this graph.
Matrix Powers and Walks
Class Work
Given 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?
Homework
Given 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?
Pre-Calc Matrices
~8~
NJCTL.org
Markov Chains
Class Work
170. 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.
Homework
171. 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 Review
Multiple Choice
1. Given the matrices at right, what are the dimensions of A?
a. 3x3
b. 2x3
c. 3x2
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
3. What element is 4(A1,2)
a. 3
b. 4
c. 6
d. 8
Pre-Calc Matrices
~9~
NJCTL.org
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. 7
c. -9
d. not possible
5. In D – B, what is the element in 2,3 position?
a. -4
b. -1
c. 1
d. 4
6. In A*E, what is the element in 1,1 position?
a. 6
b. 10
c. 24
d. not possible
7. |C|=
a. -5
b. -3
c. 3
d. 5
8. 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?
3 2
a. (
)
4 0
1 6
b. (
)
1 5
4 2
c. (
)
10 5
6 −3
d. (
)
8 4
1 2 22
10. Given (0 1 3|4) , find x, y, and z.
0 0 13
a. (2, 4, 3)
b. (6, -5, 3)
c. (-4, 1, 3)
d. cannot be determined
Pre-Calc Matrices
~10~
NJCTL.org
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. 3
c. 4
d. 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. 4
b. 7
c. 28
d. 56
14. How many ways are there to go from b1 to b2 of length 6?
a. 30
b. 15625
c. 676587
d. 761684
Extended Response
1. 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.
Pre-Calc Matrices
~11~
NJCTL.org
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 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.
Answers
3
6 −3
23. ( 2 −7 10 )
−4 6 −6
24. not possible
25. not possible
1 5
9
26. (0 −7 12 )
5 4 −9
3 1 1
27. ( 0 3 −4)
−7 0 −3
−1 0 −13
28. ( 2 −3
2 )
−2 2
6
29. not possible
8
3
−10
30. ( 2
−24 −10)
−13
2
−3
−27 0
31. ( 3
9)
−21 12
8 −8 18
32. (4 6 14)
2 0 16
33. not possible
5 −1 15
34. (4 12 7 )
4 4 13
35. not possible
1. 2 rows, 2 columns
2. 1 row, 3 columns
3. 3 rows, 2 columns
4. 4 rows, 3 columns
5. 5 rows, 1 column
6. 2 rows, 4 columns
7. -1
8. 3
9. 3
10. 6
11. 5 rows, 3 columns
12. 3rows, 1 column
13. 3 rows, 4 columns
14. 3 rows, 3 columns
15. 2 rows, 6 columns
16. 3 rows,7 columns
17. 9
18. 5
19. 0
20. -1
3
9
15
21. ( 0 −6 12 )
−3 6 −18
2
6
−16
22. ( 4 −10 12 )
−6
8
0
Pre-Calc Matrices
~12~
NJCTL.org
3 −2 14
36. (11 −2 0 )
2
2
3
−5 6
−4
37. ( 7 −8 −14)
0
2 −13
2
1
1
38. (−7 14 7 )
2
2 10
39. not possible
−17 28 −9
40. ( 23 −9 −49)
5
14 −37
41. yes, 2x4
42. no
43. yes, 1x1
44. yes, 2x9
45. yes, 7x4
14 34
46. (
)
2
6
−1 −4
47. (
)
13 −8
48. not possible
16 21 −5
49. (10 12 −2)
19 21 −2
8
0
−7
50. ( 3 −5 16 )
19 20 −22
51. no
52. yes, 4x4
53. yes, 1x3
54. no
55. yes, 4x1
−12 13
56. (
)
4
−9
−5 8
57. (
)
−36 8
58. (32)
4 8 12
59. (5 10 15)
6 12 18
−4
5 10
60. (−24 14 8 )
6
0 −2
61. 14
62. -17
63. 48
64. 1
65. -2
66. 12
67. 70
68. 7
69. -6
70. -50
71. -14
72. 11
73. -8
74. 0
75. 13
76. -10
77. -7
78. -12
79. -31
80. -6
−.1 . 3
81. (
)
. 4 −.2
4 −5
82. (
)
−3 4
83.
−1
( 31
3
84. not possible, det=0
−2 3
85. (
)
3 −4
1.5 −2
86. (
)
−.5 3
87. not possible, not square
88.
10
11
−8
11
−9
( 11
−1
11
3
11
2
11
−4
7
5
7
3
7
−2
7
−4
11
1
11
8
11 )
−2 1
0
89. ( 4 −2 . 5 )
−1 1 −.5
90.
0
(
1.5
91. (
−.5
1.5
92. (
−1
−1
93. ( −4
3
Pre-Calc Matrices
2
9
1)
9
~13~
0
−1
7
−1
14
1
2)
−2
)
1
−2.5
)
2
1
5)
3
NJCTL.org
94. not possible, det=0
−3 2.5
95. (
)
2 −1.5
−7 −5
96. (
)
−4 −3
97. not possible, not square
98.
−19
6
7
12
−3
( 4
−11
6
5
12
−1
4
124.
125.
126.
127.
128.
129.
130.
14
7
6
−1
12
1
4)
7 −3 −3
99. (−1 1
0)
−1 0
1
−24 18
5
100. ( 20 −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.
116.
117.
118.
119.
or e7
120.
131.
132.
133.
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
144.
145.
146.
147.
148.
149.
150.
151.
152.
153.
154.
155.
156.
157.
Answers will vary
E2
E1 ||e5; e4||e7
No
Need to eliminate e2, e, or e5, and e4
V1: 3; V2: 4; V3: 4; V4: 4; network: 14
0021
0101
121. [
]
2002
1120
122. 1140 𝑐𝑜𝑜𝑘𝑖𝑒𝑠
123. No, not possible to have an odd
number of odd vertices
Pre-Calc Matrices
~14~
Answers will vary
Answers will vary
E2
E1 || e3||e5; e4||e7
Yes, V5
No loops, no parallel
V1:2; V2:4; V3:5; V4:3; V5:0; network:
10000
00310
03020
01200
[0 0 0 0 0]
1050 𝑐𝑜𝑜𝑘𝑖𝑒𝑠
Yes
Answers will vary
A  eEcC
BdDfE
Yes
V=5; E=8 F=5: 5-8+5=2
Yes
Answers will vary
Already is one
BeEfAaD
AhDdCcBeE
No more than 2 odd vertices
V=5; E=8; F=5: f-8+5=2
Yes
b, c, or do
all vertices need to be even
v=6; E=6; F=2: 6-6+2=2
16
11
70
96
796
43
26
1789
NJCTL.org
158.
159.
160.
161.
162.
163.
164.
165.
166.
167.
168.
169.
1542
387,049
9
4
18
82
333
36
38
435
506
764,268
. 25 .25 .5
170. A. [ . 3 .3 .4 ]
. 7 .3 0
B. .25 .25
J
M .3
.5 .3
.7
G .4 .3
C. 22.5%’
D. J 40
. 65 .35
171. A. [
]
. 25 .75
b. .25
.65 E
.35
EE .75
c. 39%
d. 58
Pre-Calc Matrices
~15~
NJCTL.org