New Jersey Center for Teaching and Learning. Progressive Mathematics Initiative

Transcription

1 Slide 1 / 192 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website:

2 Slide 2 / 192 Pre-Calculus Matrices

3 Slide 3 / 192 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

4 Slide 4 / 192 Table of Content Circuits Definition Properties Euler Matrix Powers and Walks Markov Chains

5 Slide 5 / 192 Introduction to Matrices Return to Table of Contents

6 Slide 6 / 192 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.

7 Slide 7 / 192 What are the dimensions of the following matrices?

8 Slide 8 / How many rows does the following matrix have?

9 Slide 9 / How many columns does the following matrix have?

10 Slide 10 / How many rows does the following matrix have?

11 Slide 11 / How many columns does the following matrix have?

12 Slide 12 / How many rows does the following matrix have?

13 Slide 13 / How many columns does the following matrix have?

14 Slide 14 / 192 Matrices can be named with a capital letter. A subscript can be used to tell the dimensions of the matrix

15 Slide 15 / 192 How many rows does each matrix have? How many columns?

16 Slide 16 / How many rows does the following matrix have?

17 Slide 17 / How many columns does the following matrix have?

18 Slide 18 / How many rows does the following matrix have?

19 Slide 19 / How many columns does the following matrix have?

20 Slide 20 / 192 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 m3,4

21 Slide 21 / 192

22 Slide 22 / Identify the number in the given position.

23 Slide 23 / Identify the number in the given position.

24 Slide 24 / Identify the number in the given position.

25 Slide 25 / Identify the number in the given position.

26 Slide 26 / 192 Matrix Arithmetic Return to Table of Contents

27 Slide 27 / 192 Scalar Multiplication Return to Table of Contents

28 Slide 28 / 192 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.

29 Slide 29 / 192 Try These

30 Slide 30 / 192 Given: Let B = 6A, find b 1,2 find 6A -6

31 Slide 31 / Find the given element.

32 Slide 32 / Find the given element.

33 Slide 33 / Find the given element.

34 Slide 34 / Find the given element.

35 Slide 35 / 192 Addition Return to Table of Contents

36 Slide 36 / 192 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.

37 Slide 37 / 192 After checking to see addition is possible, add the corresponding elements.

38 Slide 38 / 192

39 Slide 39 / Add the following matrices and find the given element.

40 Slide 40 / Add the following matrices and find the given element.

41 Slide 41 / Add the following matrices and find the given element.

42 Slide 42 / Add the following matrices and find the given element.

43 Slide 43 / 192 Subtraction Return to Table of Contents

44 Slide 44 / 192 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.

45 Slide 45 / 192

46 Slide 46 / Subtract the following matrices and find the given element.

47 Slide 47 / Subtract the following matrices and find the given element.

48 Slide 48 / Subtract the following matrices and find the given element.

49 Slide 49 / Subtract the following matrices and find the given element.

50 Slide 50 / 192

51 Slide 51 / Perform the following operations on the given matrices and find the given element.

52 Slide 52 / Perform the following operations on the given matrices and find the given element.

53 Slide 53 / Perform the following operations on the given matrices and find the given element.

54 Slide 54 / Perform the following operations on the given matrices and find the given element.

55 Slide 55 / 192 Multiplication Return to Table of Contents

56 Slide 56 / 192 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.

57 Slide 57 / 192 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?

58 Slide 58 / Can the given matrices be multiplied and if so,what size will the matrix of their product be? A yes, 3x3 B yes, 4x4 C yes, 3x4 D they cannot be multiplied

59 Slide 59 / Can the given matrices be multiplied and if so,what size will the matrix of their product be? A yes, 3x3 B yes, 4x4 C yes, 3x4 D they cannot be multiplied

60 Slide 60 / Can the given matrices be multiplied and if so,what size will the matrix of their product be? A yes, 3x3 B yes, 4x4 C yes, 3x4 D they cannot be multiplied

61 Slide 61 / Can the given matrices be multiplied and if so,what size will the matrix of their product be? A yes, 3x3 B yes, 4x4 C yes, 3x4 D they cannot be multiplied

62 Slide 62 / 192 To multiply matrices, distribute the rows of first to the columns of the second. Add the products.

63 Slide 63 / 192 Try These

64 Slide 64 / 192 Try These

65 Slide 65 / 192 Try These

66 Slide 66 / Perform the following operations on the given matrices and find the given element.

67 Slide 67 / Perform the following operations on the given matrices and find the given element.

68 Slide 68 / Perform the following operations on the given matrices and find the given element.

69 Slide 69 / Perform the following operations on the given matrices and find the given element.

70 Slide 70 / 192 Solving Systems of Equations using Matrices Return to Table of Contents

71 Slide 71 / 192 Finding Determinants of 2x2 & 3x3 Return to Table of Contents

72 Slide 72 / 192 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.

73 Slide 73 / 192 To find the determinant of a 2x2 matrix: The product of the primary diagonal minus the product of the secondary diagonal. Example:

74 Slide 74 / 192 Try These:

75 Slide 75 / Find the determinant of the following:

76 Slide 76 / Find the determinant of the following:

77 Slide 77 / Find the determinant of the following:

78 Slide 78 / Find the determinant of the following:

79 Slide 79 / 192 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.

80 Slide 80 / 192

81 Slide 81 / 192 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve.

82 Slide 82 / 192 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve.

83 Slide 83 / 192 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve.

84 Slide 84 / 192 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 Slide 85 / 192 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.

86 Slide 86 / Find the determinant of the following:

87 Slide 87 / Find the determinant of the following:

88 Slide 88 / Find the determinant of the following:

89 Slide 89 / Find the determinant of the following:

90 Slide 90 / 192 Finding the Inverse of 2x2 & 3x3 Return to Table of Contents

91 Slide 91 / 192 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:

92 Slide 92 / 192 Property of the IdentityMatrix

93 Slide 93 / 192 The inverse of matrix A is matrix A-1. The product of a matrix and its inverse is the identity matrix, I. example:

94 Slide 94 / 192 Note: Not all matrices have an inverse. matrix must be square the determinant of the matrix cannot = 0

95 Slide 95 / 192 Finding the inverse of a 2x2 matrix Example: Find the inverse of matrix M.

96 Slide 96 / 192 check:

97 Slide 97 / 192 Find the inverse of matrix A

98 Slide 98 / 192 Find the inverse of matrix A

99 Slide 99 / 192 Find the inverse of matrix A

100 Slide 100 / 192 Find the inverse of matrix A

101 Slide 101 / 192 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.

102 Slide 102 / 192 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.

103 Slide 103 / 192 Now we know the rules, let's play. Beginning matrix Subtracted 2 times row 1 from row 2 Switched rows 1&2 Subtracted 6 times row 1 from row 3 Divided row 1 by 4 Switched rows 2&3

104 Slide 104 / 192 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 times row 3 from row 2 Sub times row 3 from row 1

105 Slide 105 / 192 We began with this: We ended with this: Meaning the inverse of is

106 Slide 106 / 192 Find the inverse of:

107 Slide 107 / 192 Find the inverse of:

108 Slide 108 / 192 Representing 2- and 3Variable Systems Return to Table of Contents

109 Slide 109 / 192 Solving Matrix Equations Return to Table of Contents

110 Slide 110 / 192 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

111 Slide 111 / 192 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:

112 Slide 112 / 192 Solve: Step 1: Step 2: find the inverse of

113 Slide 113 / 192 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 = 7.

114 Slide 114 / 192 Rewrite each system as a product of matrices.

115 Slide 115 / 192 Find x and y

116 Slide 116 / 192 Find x and y

117 Slide 117 / Is this system ready to be made into a matrix equation? Yes No

118 Slide 118 / Which of the following is the correct matrix equation for the system? A C B D

119 Slide 119 / What is the determinant of: A -17 B -13 C 13 D 17

120 Slide 120 / What is the inverse of: A B C D

121 Slide 121 / Find the solution to What is the x-value?

122 Slide 122 / Find the solution to What is the y-value?

123 Slide 123 / Is this system ready to be made into a matrix equation? Yes No

124 Slide 124 / Which of the following is the correct matrix equation for the system? A C B D

125 Slide 125 / What is the determinant of: A -10 B -2 C 2 D 10

126 Slide 126 / What is the inverse of: A B C D

127 Slide 127 / Find the solution to What is the x-value?

128 Slide 128 / Find the solution to What is the y-value?

129 Slide 129 / 192 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.

130 Slide 130 / 192 Start Swapped row 2 and 3 (rather divide by 3 than 7) Swap Rows 1&2 Subtract 5 times row 1 from row 2 Subtract row 1 from row 2 Divide row 2 by -3 Add 7 times row 2 to row 3 Subtract 2 times row 2 from row 1

131 Slide 131 / 192 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 = 2.

132 Slide 132 / 192 Convert the system to an augmented matrice. Solve using row reduction

133 Slide 133 / 192 Convert the system to an augmented matrice. Solve using row reduction

134 Slide 134 / 192 Convert the system to an augmented matrice. Solve using row reduction

135 Slide 135 / 192 Circuits Return to Table of Contents

136 Slide 136 / 192 Definition Return to Table of Contents

137 Slide 137 / 192 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.

138 Slide 138 / 192 Edge endpoints

139 Slide 139 / 192 Vocab Adjacent edges share a vertex. Adjacent vertices are connected by an edge. e5 and e6 are parallel because they connect the same vertices. A e1 and e7 are loops. v8 is isolated because it is not the endpoint for any edges. A simple graph has no loops and no parallel edges.

140 Slide 140 / 192 Make a simple graph with vertices {a, b, c, d} and as many edges as possible.

141 Slide 141 / Which edge(s) are loops? A e1 G v1 e2 H v2 B C e3 I v3 D e4 J E e5 F e6 v4

142 Slide 142 / Which edge(s) are parallel? A e1 G v1 e2 H v2 B C e3 I v3 D e4 J E e5 F e6 v4

143 Slide 143 / Which edge(s) are adjacent to e4? A e1 G v1 e2 H v2 B C e3 I v3 D e4 J E e5 F e6 v4

144 Slide 144 / 192 Which vertices are adjacent to v4? 62 A e1 G v1 e2 H v2 B C e3 I v3 D e4 J E e5 F e6 v4

145 Slide 145 / 192 Which vertex is isolated? 63 A e1 G v1 e2 H v2 B C e3 I v3 D e4 J E e5 F none v4

146 Slide 146 / 192 Some graphs will show that an edge can be traversed in only one direction, like one way streets. This is a directed graph.

147 Slide 147 / 192 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 v4 to v5.

148 Slide 148 / How many paths are there from v2 to v3?

149 Slide 149 / Which vertex is isolated?

150 Slide 150 / 192 Properties Return to Table of Contents

151 Slide 151 / 192 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.

152 Slide 152 / 192 Complete Graph The number of edges of a complete graph is

153 Slide 153 / The Duggers, who are huggers, had a family reunion. 50 family members attended. How many hugs were exchanged?

154 Slide 154 / 192 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?

155 Slide 155 / What is the degree of A? A C B

156 Slide 156 / What is the degree of B? A C B

157 Slide 157 / What is the degree of C? A C B

158 Slide 158 / What is the degree of the network? A C B

159 Slide 159 / 192 Corollaries: the degree of a network is even a network will have an even number of odd vertices

160 Slide 160 / 192 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? An odd number of handshakes means what is the degree of those verticces? Corollaries: the degree of a network is even a network will have an even number of odd vertices

161 Slide 161 / 192 Euler Return to Table of Contents

162 Slide 162 / 192 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?

163 Slide 163 / 192 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.

164 Slide 164 / 192 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.

165 Slide 165 / 192 Euler determined that it was not possible because there are 4 odd vertices.

166 Slide 166 / 192 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.

167 Slide 167 / 192 For a network to be an Euler circuit, every vertex has an even degree.

168 Slide 168 / Which is a walk from v1 to v5? A v1,e3,v3,e4, v5 B v1,e2,v2,e3,v3,e5,v4,e7,v5 e4 C v1,e3,e2,e7,v5 D v1, e3,v3,e5,v4,e7,v5 e3 v3 e5 v1 e1 v4 e2 v2 e7 v5 e8

169 Slide 169 / Is this graph traversable? Yes No e3 v3 e4 e5 v1 e1 v4 v2 e7 v5 e8

170 Slide 170 / 192 Connected vertices have at least on walk connecting them. e3 v3 e4 e5 v1 e1 v4 v2 e7 v5 e8 Connected graphs have all connected vertices

171 Slide 171 / 192 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 = 2 Tetrahedron =2

172 Slide 172 / 192 Apply Euler's Formula to circuits. Add 1 to faces for the not enclosed region. V=5 E=7 F=3+1 V=7 E=9 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

173 Slide 173 / How many 'faces' does this graph have?

174 Slide 174 / How many 'edges' does this graph have?

175 Slide 175 / How many 'vertices' does this graph have?

176 Slide 176 / For this graph, what does V - E + F=?

177 Slide 177 / 192 Matrix Powers and Walks Return to Table of Contents

178 Slide 178 / 192 Earlier in this unit, we looked at adjacency matrices for directed graphs.

179 Slide 179 / 192 There are also adjacency matrices for undirected graphs. a1 a4 a2 main diagonal a3 What do the numbers on the main diagonal represent? What can be said about the halves of adjacency matrix?

180 Slide 180 / 192 The number of walks of length 1 from a1 to a3 is 3. How many walks of length 2 are there from a1 to a3? By raising the matrix to the power of the desired length walk, the element in the 1st row 3rd column is the answer. a1 a4 a2 a3 Why does this work? When multiplying, its the 1st row, all the walks length one from a1, by column 3, all the walks length 1 from a 3.

181 Slide 181 / How many walks of length 2 are there from a2 to a4? a1 a4 a2 a3

182 Slide 182 / How many walks of length 3 are there from a2 to a2? a1 a4 a2 a3

183 Slide 183 / How many walks of length 5 are there from a1 to a3? a1 a4 a2 a3

184 Slide 184 / 192 Markov Chains Return to Table of Contents

185 Slide 185 / 192 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..40 I.60 < <.40< < <.30 C < <.10 <.10 O <.50 <

186 Slide 186 / 192 What is the probability that a car commercial follows an Internet commercial?.40 I.60 < <.40< < <.30 C < <.10 <.10 O <.50 <

187 Slide 187 / 192 An adjacency matrix that shows the probabilities of what happens next is called a transition matrix..40 I.60 < <.40< < <.30 C < <.10 <.10 O <.50 Since each row adds to 1 (100%) this matrix is also called a stochastic matrix. <

188 Slide 188 / 192 What will the commercial be 2 commercials after a car ad? Using the properties from walks, square the transition matrix..40 I.60 < <.40< < <.30 C < <.10 <.10 O <.50 The first row gives the likelihood of the type of ad following a car ad. <

189 Slide 189 / 192 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. <

190 Slide 190 / 192 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.

191 Slide 191 / Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born in 10 generations?

192 Slide 192 / Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born to non-champions in 2 generations?