Pre Calculus. Matrices.
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1 1
2 Pre Calculus Matrices
3 Table of Content Introduction to Matrices Matrix Arithmetic Scalar Multiplication Addition Subtraction Multiplication Solving Systems of Equations using Matrices Finding Determinants of 2x2 & 3x3 Finding the Inverse of 2x2 & 3x3 Representing 2 and 3 variable systems Solving Matrix Equations Circuits 3
4 Table of Content Circuits Definition Properties Euler Matrix Powers and Walks Markov Chains 4
5 Introduction to Matrices Return to Table of Contents 5
6 A matrix is an ordered array. The matrix consists of rows and columns. Columns Rows This matrix has 3 rows and 3 columns, it is said to be 3x3. 6
7 What are the dimensions of the following matrices? 7
8 1 How many rows does the following matrix have? 8
9 2 How many columns does the following matrix have? 9
10 3 How many rows does the following matrix have? 10
11 4 How many columns does the following matrix have? 11
12 5 How many rows does the following matrix have? 12
13 6 How many columns does the following matrix have? 13
14 Matrices can be named with a capital letter. A subscript can be used to tell the dimensions of the matrix 14
15 How many rows does each matrix have? How many columns? 15
16 7 How many rows does the following matrix have? 16
17 8 How many columns does the following matrix have? 17
18 9 How many rows does the following matrix have? 18
19 10 How many columns does the following matrix have? 19
20 We can find an entry in a certain position of a matrix. To find the number in the third row,fourth column of matrix M write m 3,4 20
21 21
22 11 Identify the number in the given position. 22
23 12 Identify the number in the given position. 23
24 13 Identify the number in the given position. 24
25 14 Identify the number in the given position. 25
26 Matrix Arithmetic Return to Table of Contents 26
27 Scalar Multiplication Return to Table of Contents 27
28 A scalar multiple is when a single number is multiplied to the entire matrix. To multiply by a scalar, distribute the number to each entry in the matrix. 28
29 Try These 29
30 Given: find 6A Answer Let B = 6A, find b 1,2 30
31 15 Find the given element. 31
32 16 Find the given element. 32
33 17 Find the given element. 33
34 18 Find the given element. 34
35 Addition Return to Table of Contents 35
36 To add matrices, they must have the same dimensions. That is, the same number of rows, same number of columns. Given: State whether the following addition problems are possible or not possible. 36
37 After checking to see addition is possible, add the corresponding elements. 37
38 38
39 19 Add the following matrices and find the given element. 39
40 20 Add the following matrices and find the given element. 40
41 21 Add the following matrices and find the given element. 41
42 22 Add the following matrices and find the given element. 42
43 Subtraction Return to Table of Contents 43
44 To be able to subtract matrices, they must have the same dimensions, like addition. Method 1: Subtract corresponding elements. Method 2: Change to addition with a negative scalar. Note: Method 2 adds a step but less likely to have a sign error. 44
45 45
46 23 Subtract the following matrices and find the given element. 46
47 24 Subtract the following matrices and find the given element. 47
48 25 Subtract the following matrices and find the given element. 48
49 26 Subtract the following matrices and find the given element. 49
50 50
51 27 Perform the following operations on the given matrices and find the given element. 51
52 28 Perform the following operations on the given matrices and find the given element. 52
53 29 Perform the following operations on the given matrices and find the given element. 53
54 30 Perform the following operations on the given matrices and find the given element. 54
55 Multiplication Return to Table of Contents 55
56 Multiplication, like addition, not all matrices can be multiplied. The number of columns in the first matrix has to be the same as the number of rows in the second matrix. 56
57 State whether each pair of matrices can be multiplied, if so what will the dimensions of the their product be? Compare the answers from column 1 to column 2: Does AB=BA? Conclusions? 57
58 31 Can the given matrices be multiplied and if so,what size will the matrix of their product be? A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied 58
59 32 Can the given matrices be multiplied and if so,what size will the matrix of their product be? A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied 59
60 33 Can the given matrices be multiplied and if so,what size will the matrix of their product be? A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied 60
61 34 Can the given matrices be multiplied and if so,what size will the matrix of their product be? A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied 61
62 To multiply matrices, distribute the rows of first to the columns of the second. Add the products. 62
63 Try These 63
64 Try These 64
65 Try These 65
66 35 Perform the following operations on the given matrices and find the given element. 66
67 36 Perform the following operations on the given matrices and find the given element. 67
68 37 Perform the following operations on the given matrices and find the given element. 68
69 38 Perform the following operations on the given matrices and find the given element. 69
70 Solving Systems of Equations using Matrices Return to Table of Contents 70
71 Finding Determinants of 2x2 & 3x3 Return to Table of Contents 71
72 A determinant is a value assigned to a square matrix. This value is used as scale factor for transformations of matrices. The bars for determinant look like absolute value signs but are not. 72
73 To find the determinant of a 2x2 matrix: The product of the primary diagonal minus the product of the secondary diagonal. Example: 73
74 Try These: 74
75 39 Find the determinant of the following: 75
76 40 Find the determinant of the following: 76
77 41 Find the determinant of the following: 77
78 42 Find the determinant of the following: 78
79 Finding the Determinant of a 3x3 Matrix Use the first row of the matrix to expand the 3x3 to 3 2x2 matrices, then use the 2x2 method. Eliminate the both the row and column the 1 is in. Eliminate the both the row and column the 2 is in. Eliminate the both the row and column the 3 is in. The second number is subtracted. Had 2 been a negative then this would subtracting a negative. 79
80 80
81 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. 81
82 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. 82
83 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. 83
84 Begin the expansion by rewriting the determinant 3 times with the first row with the coefficients. Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. 84
85 Begin the expansion by rewriting the determinant 3 times with the first row with the coefficients. Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. 85
86 43 Find the determinant of the following: 86
87 44 Find the determinant of the following: 87
88 45 Find the determinant of the following: 88
89 46 Find the determinant of the following: 89
90 Finding the Inverse of 2x2 & 3x3 Return to Table of Contents 90
91 The Identity Matrix ( I ) is a square matrix with 1's on its primary diagonal and 0's as the other elements. 2x2 Identity Matrix: 3x3 Identity Matrix: 4x4 Identity Matrix: 91
92 Property of the IdentityMatrix 92
93 The inverse of matrix A is matrix A 1. The product of a matrix and its inverse is the identity matrix, I. example: 93
94 Note: Not all matrices have an inverse. matrix must be square the determinant of the matrix cannot = 0 94
95 Finding the inverse of a 2x2 matrix Example: Find the inverse of matrix M. 95
96 check: 96
97 Find the inverse of matrix A 97
98 Find the inverse of matrix A 98
99 Find the inverse of matrix A 99
100 Find the inverse of matrix A 100
101 Inverse of a 3x3 Matrix This technique involves creating an Augmented Matrix to start. Matrix we want the inverse of. Identity Matrix Note: This technique can be done for any size square matrix. 101
102 Inverse of a 3x3 Matrix Think of this technique, Row Reduction, as a number puzzle. Goal: Reduce the left hand matrix to the identity matrix. Rules: the entire row stays together, what ever is done to an element of a row is done to the entire row allowed to switch any row with any other row may divide/multiply the entire row by a non zero number adding/subtracting one entire row from another is permitted Caution: Not all square matrices are invertible, if a row on the left goes to all zeros there is no inverse. 102
103 Now we know the rules, let's play. Beginning matrix Subtracted 2 times row 1 from row 2 Switched rows 1&2 Divided row 1 by 4 Subtracted 6 times row 1 from row 3 Switched rows 2&3 103
104 Cont. from previous slide Div row 2 by 4 Div row 3 by 4.5 Sub 1.5 times row 2 from row 1 Sub.625 times row 3 from row 2 Sub times row 3 from row 1 104
105 We began with this: We ended with this: Meaning the inverse of is 105
106 Find the inverse of: 106
107 Find the inverse of: 107
108 Representing 2 and 3 Variable Systems Return to Table of Contents 108
109 Solving Matrix Equations Return to Table of Contents 109
110 Matrices can be used to solve systems of equations. Consider the system of equations: Note: equations need to be in standard form. Rewrite the system into a product of matrices: coefficients variables constants 110
111 To solve this equation, you need to isolate the variables, but how? The inverse of the coefficient matrix multiplied to both sides will work. Think of it as: 111
112 Solve: Step 1: Step 2: find the inverse of 112
113 Step 3: Recall that in matrix multiplication, the commutative property doesn't hold true. The associative property does work: (AB)C=A(BC) The solution to the system is x = 3 and y =
114 Rewrite each system as a product of matrices. 114
115 Find x and y 115
116 Find x and y 116
117 47 Is this system ready to be made into a matrix equation? Yes No 117
118 48 Which of the following is the correct matrix equation for the system? A C B D 118
119 49 What is the determinant of: A 17 B 13 C 13 D
120 50 What is the inverse of: A B C D 120
121 51 Find the solution to What is the x value? 121
122 52 Find the solution to What is the y value? 122
123 53 Is this system ready to be made into a matrix equation? Yes No 123
124 54 Which of the following is the correct matrix equation for the system? A C B D 124
125 55 What is the determinant of: A 10 B 2 C 2 D
126 56 What is the inverse of: A B C D 126
127 57 Find the solution to What is the x value? 127
128 58 Find the solution to What is the y value? 128
129 For systems of equations with 3 or more variables, create an augmented matrices with the coefficients on one side and the constants on the other. Row reduce. When the identity matrix is on the left, the solutions are on the right. 129
130 Start Swapped row 2 and 3 (rather divide by 3 than 7) Swap Rows 1&2 Divide row 2 by 3 Subtract 5 times row 1 from row 2 Subtract row 1 from row 2 Add 7 times row 2 to row 3 Subtract 2 times row 2 from row 1 130
131 From Previous slide Divide row 3 by 37/3 Subtract 2/3 times row 3 from row 2 Subtract 5/3 times row 3 from row 1 The solution to the system is x = 1, y = 1, and z =
132 Convert the system to an augmented matrice. Solve using row reduction 132
133 Convert the system to an augmented matrice. Solve using row reduction 133
134 Convert the system to an augmented matrice. Solve using row reduction 134
135 Circuits Return to Table of Contents 135
136 Definition Return to Table of Contents 136
137 A Graph of a network consists of vertices (points) and edges (edges connect the points) The points marked v are the vertices, or nodes, of the network. The edges are e. 137
138 Edge endpoints 138
139 Vocab Adjacent edges share a vertex. Adjacent vertices are connected by an edge. e 5 and e 6 are parallel because they connect the same vertices. A e 1 and e 7 are loops. v 8 is isolated because it is not the endpoint for any edges. A simple graph has no loops and no parallel edges. 139
140 Make a simple graph with vertices {a, b, c, d} and as many edges as possible. 140
141 59 Which edge(s) are loops? A e 1 B e 2 C e 3 D e 4 G v 1 H v 2 I v 3 J v 4 E e 5 F e 6 141
142 60 Which edge(s) are parallel? A e 1 B e 2 C e 3 D e 4 G v 1 H v 2 I v 3 J v 4 E e 5 F e 6 142
143 61 Which edge(s) are adjacent to e 4? A e 1 B e 2 C e 3 D e 4 G v 1 H v 2 I v 3 J v 4 E e 5 F e 6 143
144 62 Which vertices are adjacent to v 4? A e 1 B e 2 C e 3 D e 4 G v 1 H v 2 I v 3 J v 4 E e 5 F e 6 144
145 63 Which vertex is isolated? A e 1 B e 2 C e 3 D e 4 G v 1 H v 2 I v 3 J v 4 E e 5 F none 145
146 Some graphs will show that an edge can be traversed in only one direction, like one way streets. This is a directed graph. 146
147 An adjacency matrix shows the number of paths from one vertex to another. So row 4 column 5 shows that there is 1 path from v 4 to v
148 64 How many paths are there from v 2 to v 3? 148
149 65 Which vertex is isolated? 149
150 Properties Return to Table of Contents 150
151 Complete Graph Every vertex is connected to every other by one edge. So at a meeting with 8 people, each person shook hands with every other person once. The graph shows the handshakes. So all 8 people shook hands 7 times, that would seem like 56 handshakes. But there 28 edges to the graph. Person A shaking with B and B shaking with A is the same handshake. 151
152 Complete Graph The number of edges of a complete graph is 152
153 66 The Duggers, who are huggers, had a family reunion. 50 family members attended. How many hugs were exchanged? 153
154 Degrees The degree of a vertex is the number edges that have the vertex as an endpoint. Loops count as 2. The degree of a network is the sum of the degrees of the vertices. The degree of the network is twice the number of edges. Why? 154
155 67 What is the degree of A? A C B 155
156 68 What is the degree of B? A C B 156
157 69 What is the degree of C? A C B 157
158 70 What is the degree of the network? A C B 158
159 Corollaries: the degree of a network is even a network will have an even number of odd vertices 159
160 Can odd number of people at a party shake hands with an odd number of people? Think about the corollaries. An odd number of people means how many vertices? Corollaries: the degree of a network is even a network will have an even number of odd vertices An odd number of handshakes means what is the degree of those verticces? 160
161 Euler Return to Table of Contents 161
162 Konisberg Bridge Problem Konisberg was a city in East Prussia, built on the banks of the Pregol River. In the middle of the river are 2 islands, connected to each other and the banks by a series of bridges. The Konisberg Bridge Problem asks if it is possible to travel each bridge exactly once and end up back where you started? 162
163 In 1736, 19 year old Leonhard Euler, one of the greatest mathematicians of all time, solve the problem. Euler, made a graph of the city with the banks and islands as vertices and the bridges as edges. He then developed rules about traversable graphs. 163
164 Traversable A network is traversable if each edge can be traveled travelled exactly once. In this puzzle, you are asked to draw the house,or envelope, without repeating any lines. Determine the degree of each vertex. Traversable networks will have 0 or 2 odd vertices. If there are 2 odd vertices start at one and end at the other. 164
165 Euler determined that it was not possible because there are 4 odd vertices. 165
166 A walk is a sequence of edges and vertices from a to b. A path is a walk with no edge repeated.(traversable) A circuit is a path that starts and stops at the same vertex. An Euler circuit is a circuit that can start at any vertex. 166
167 For a network to be an Euler circuit, every vertex has an even degree. 167
168 71 Which is a walk from v 1 to v 5? A v 1,e 3,v 3,e 4, v 5 B v 1,e 2,v 2,e 3,v 3,e 5,v 4,e 7,v 5 v 3 v 1 e 2 C v 1,e 3,e 2,e 7,v 5 D v 1, e 3,v 3,e 5,v 4,e 7,v 5 e 4 e 3 e 1 e5 v 4 e 7 v 2 v 5 e 8 168
169 72 Is this graph traversable? Yes No v 3 v 1 e 4 e 3 e 1 e5 v 4 v 2 e 7 v 5 e 8 169
170 Connected vertices have at least on walk connecting them. v 3 v 1 e 4 e 3 e 1 e5 v 4 v 2 e 7 v 5 e 8 Connected graphs have all connected vertices 170
171 For all Polyhedra, Euler's Formula V E + F = 2 V is the number vertices E is the number of edges F is the number of faces Pentagonal Prism Tetrahedron = =2 171
172 Apply Euler's Formula to circuits. Add 1 to faces for the not enclosed region. V=5 E=7 F=3+1 Euler's Formula V E + F = 2 V is the number vertices E is the number of edges F is the number of faces V=7 E=9 F=
173 73 How many 'faces' does this graph have? 173
174 74 How many 'edges' does this graph have? 174
175 75 How many 'vertices' does this graph have? 175
176 76 For this graph, what does V E + F=? 176
177 Matrix Powers and Walks Return to Table of Contents 177
178 Earlier in this unit, we looked at adjacency matrices for directed graphs. 178
179 There are also adjacency matrices for undirected graphs. a 1 a 2 a 4 main diagonal What do the numbers on the main diagonal represent? a 3 What can be said about the halves of adjacency matrix? 179
180 The number of walks of length 1 from a 1 to a 3 is 3. a 1 How many walks of length 2 are there from a 1 to a 3? a 2 a 4 By raising the matrix to the power of the desired length walk, the element in the 1st row 3rd column is the answer. a 3 Why does this work? When multiplying, its the 1st row, all the walks length one from a 1, by column 3, all the walks length 1 from a
181 77 How many walks of length 2 are there from a 2 to a 4? a 1 a 2 a 4 a 3 181
182 78 How many walks of length 3 are there from a 2 to a 2? a 1 a 2 a 4 a 3 182
183 79 How many walks of length 5 are there from a 1 to a 3? a 1 a 2 a 4 a 3 183
184 Markov Chains Return to Table of Contents 184
185 During the Super Bowl, it was determined that the commercials could be divided into 3 categories: car, Internet sites, and other. The directed graph below shows the probability that after a commercial aired what the probability for the next type of commercial..10 < C <.40 <.40 < <.40 < <.10 I.20 < < O.50 < 185
186 What is the probability that a car commercial follows an Internet commercial?.10 < C <.40 <.40 < <.40 < <.10 I.20 < < O.50 < 186
187 .10 < An adjacency matrix that shows the probabilities of what happens next is called a transition matrix..40 C.40 < < < <.40 < <.10 I.20 < < O.50 Since each row adds to 1 (100%) this matrix is also called a stochastic matrix. < 187
188 .10 < C What will the commercial be 2 commercials after a car ad? Using the properties from walks, square the transition matrix..40 <.40 < < <.40 < <.10 I.20 < < O.50 The first row gives the likelihood of the type of ad following a car ad. < 188
189 This method can be applied for any number of ads. But notice what happens to the elements as we get to 10 ads away. This means that no matter what commercial is on, there is an 18% chance that 10 ads from now will be an Internet ad. < 189
190 Horse breeders found that the if a champion horse sired an offspring it had 40% of being a champion. If a non champion horses had offspring, they were 35% likely of being champions. Make a graph and a transition matrix. 190
191 80 Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born in 10 generations? 191
192 81 Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born to non champions in 2 generations? 192
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