Slide 1 / 192 Pre-Calculus Slide 2 / 192 Matrices 2015-03-23 www.njctl.org 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 Slide 3 / 192
Table of Content Circuits Definition Properties Euler Matrix Powers and Walks Markov Chains Slide 4 / 192 Slide 5 / 192 Introduction to Matrices Return to Table of Contents A matrix is an ordered array. Slide 6 / 192 The matrix consists of rows and columns. Columns Rows This matrix has 3 rows and 3 columns, it is said to be 3x3.
What are the dimensions of the following matrices? Slide 7 / 192 1 How many rows does the following matrix have? Slide 8 / 192 2 How many columns does the following matrix have? Slide 9 / 192
3 How many rows does the following matrix have? Slide 10 / 192 4 How many columns does the following matrix have? Slide 11 / 192 5 How many rows does the following matrix have? Slide 12 / 192
6 How many columns does the following matrix have? Slide 13 / 192 Slide 14 / 192 Slide 15 / 192 How many rows does each matrix have? How many columns?
Slide 16 / 192 Slide 17 / 192 9 How many rows does the following matrix have? Slide 18 / 192
10 How many columns does the following matrix have? Slide 19 / 192 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 m 3,4 Slide 21 / 192
11 Identify the number in the given position. Slide 22 / 192 12 Identify the number in the given position. Slide 23 / 192 13 Identify the number in the given position. Slide 24 / 192
14 Identify the number in the given position. Slide 25 / 192 Slide 26 / 192 Matrix Arithmetic Return to Table of Contents Slide 27 / 192 Scalar Multiplication Return to Table of Contents
A scalar multiple is when a single number is multiplied to the entire matrix. Slide 28 / 192 To multiply by a scalar, distribute the number to each entry in the matrix. Try These Slide 29 / 192 Given: find 6A Slide 30 / 192 Let B = 6A, find b 1,2
Given: find 6A Slide 30 (Answer) / 192 Answer Let B = 6A, find b 1,2-6 [This object is a pull tab] 15 Find the given element. Slide 31 / 192 16 Find the given element. Slide 32 / 192
17 Find the given element. Slide 33 / 192 18 Find the given element. Slide 34 / 192 Slide 35 / 192 Addition Return to Table of Contents
Slide 36 / 192 After checking to see addition is possible, add the corresponding elements. Slide 37 / 192 Slide 38 / 192
19 Add the following matrices and find the given element. Slide 39 / 192 20 Add the following matrices and find the given element. Slide 40 / 192 21 Add the following matrices and find the given element. Slide 41 / 192
22 Add the following matrices and find the given element. Slide 42 / 192 Slide 43 / 192 Subtraction Return to Table of Contents To be able to subtract matrices, they must have the same dimensions, like addition. Slide 44 / 192 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.
Slide 45 / 192 23 Subtract the following matrices and find the given element. Slide 46 / 192 24 Subtract the following matrices and find the given element. Slide 47 / 192
25 Subtract the following matrices and find the given element. Slide 48 / 192 26 Subtract the following matrices and find the given element. Slide 49 / 192 Slide 50 / 192
27 Perform the following operations on the given matrices and find the given element. Slide 51 / 192 28 Perform the following operations on the given matrices and find the given element. Slide 52 / 192 29 Perform the following operations on the given matrices and find the given element. Slide 53 / 192
Slide 54 / 192 Slide 55 / 192 Multiplication Return to Table of Contents Slide 56 / 192
Slide 57 / 192 31 Can the given matrices be multiplied and if so,what size will the matrix of their product be? Slide 58 / 192 A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied 32 Can the given matrices be multiplied and if so,what size will the matrix of their product be? Slide 59 / 192 A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied
33 Can the given matrices be multiplied and if so,what size will the matrix of their product be? Slide 60 / 192 A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied 34 Can the given matrices be multiplied and if so,what size will the matrix of their product be? Slide 61 / 192 A B C D yes, 3x3 yes, 4x4 yes, 3x4 they cannot be multiplied To multiply matrices, distribute the rows of first to the columns of the second. Add the products. Slide 62 / 192
Try These Slide 63 / 192 Try These Slide 64 / 192 Slide 65 / 192
35 Perform the following operations on the given matrices and find the given element. Slide 66 / 192 36 Perform the following operations on the given matrices and find the given element. Slide 67 / 192 37 Perform the following operations on the given matrices and find the given element. Slide 68 / 192
38 Perform the following operations on the given matrices and find the given element. Slide 69 / 192 Slide 70 / 192 Solving Systems of Equations using Matrices Return to Table of Contents Slide 71 / 192 Finding Determinants of 2x2 & 3x3 Return to Table of Contents
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. Slide 73 / 192 Try These: Slide 74 / 192
39 Find the determinant of the following: Slide 75 / 192 40 Find the determinant of the following: Slide 76 / 192 41 Find the determinant of the following: Slide 77 / 192
42 Find the determinant of the following: Slide 78 / 192 Slide 79 / 192 Slide 80 / 192
Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. Slide 81 / 192 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. Slide 82 / 192 Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve. Slide 83 / 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. 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. Slide 85 / 192 43 Find the determinant of the following: Slide 86 / 192
44 Find the determinant of the following: Slide 87 / 192 45 Find the determinant of the following: Slide 88 / 192 46 Find the determinant of the following: Slide 89 / 192
Slide 90 / 192 Finding the Inverse of 2x2 & 3x3 Return to Table of Contents The Identity Matrix ( I ) is a square matrix with 1's on its primary diagonal and 0's as the other elements. Slide 91 / 192 2x2 Identity Matrix: 3x3 Identity Matrix: 4x4 Identity Matrix: Property of the IdentityMatrix Slide 92 / 192
Slide 93 / 192 Slide 94 / 192 Note: Not all matrices have an inverse. matrix must be square the determinant of the matrix cannot = 0 Slide 95 / 192
Slide 96 / 192 Find the inverse of matrix A Slide 97 / 192 Find the inverse of matrix A Slide 98 / 192
Find the inverse of matrix A Slide 99 / 192 Find the inverse of matrix A Slide 100 / 192 Inverse of a 3x3 Matrix This technique involves creating an Augmented Matrix to start. Slide 101 / 192 Matrix we want the inverse of. Identity Matrix Note: This technique can be done for any size square matrix.
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. Slide 102 / 192 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. Slide 103 / 192 Slide 104 / 192
Slide 105 / 192 We began with this: We ended with this: Meaning the inverse of is Find the inverse of: Slide 106 / 192 Find the inverse of: Slide 107 / 192
Slide 108 / 192 Representing 2- and 3- Variable Systems Return to Table of Contents Slide 109 / 192 Solving Matrix Equations Return to Table of Contents Slide 110 / 192
Slide 111 / 192 Slide 112 / 192 Slide 113 / 192
Rewrite each system as a product of matrices. Slide 114 / 192 Find x and y Slide 115 / 192 Find x and y Slide 116 / 192
47 Is this system ready to be made into a matrix equation? Slide 117 / 192 Yes No 48 Which of the following is the correct matrix equation for the system? Slide 118 / 192 A C B D 49 What is the determinant of: Slide 119 / 192 A -17 B -13 C 13 D 17
50 What is the inverse of: Slide 120 / 192 A B C D 51 Find the solution to What is the x-value? Slide 121 / 192 52 Find the solution to What is the y-value? Slide 122 / 192
53 Is this system ready to be made into a matrix equation? Slide 123 / 192 Yes No 54 Which of the following is the correct matrix equation for the system? Slide 124 / 192 A C B D 55 What is the determinant of: Slide 125 / 192 A -10 B -2 C 2 D 10
56 What is the inverse of: Slide 126 / 192 A B C D 57 Find the solution to What is the x-value? Slide 127 / 192 58 Find the solution to What is the y-value? Slide 128 / 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. Slide 129 / 192 Row reduce. When the identity matrix is on the left, the solutions are on the right. Start Swapped row 2 and 3 (rather divide by 3 than 7) Slide 130 / 192 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 From Previous slide Slide 131 / 192 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.
Convert the system to an augmented matrice. Solve using row reduction Slide 132 / 192 Convert the system to an augmented matrice. Solve using row reduction Slide 133 / 192 Convert the system to an augmented matrice. Solve using row reduction Slide 134 / 192
Slide 135 / 192 Circuits Return to Table of Contents Slide 136 / 192 Definition Return to Table of Contents A Graph of a network consists of vertices (points) and edges (edges connect the points) Slide 137 / 192 The points marked v are the vertices, or nodes, of the network. The edges are e.
Slide 138 / 192 Vocab Adjacent edges share a vertex. Slide 139 / 192 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. Make a simple graph with vertices {a, b, c, d} and as many edges as possible. Slide 140 / 192
59 Which edge(s) are loops? Slide 141 / 192 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 60 Which edge(s) are parallel? Slide 142 / 192 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 61 Which edge(s) are adjacent to e 4? Slide 143 / 192 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
62 Which vertices are adjacent to v 4? Slide 144 / 192 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 63 Which vertex is isolated? Slide 145 / 192 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 Some graphs will show that an edge can be traversed in only one direction, like one way streets. Slide 146 / 192 This is a directed graph.
Slide 147 / 192 64 How many paths are there from v 2 to v 3? Slide 148 / 192 65 Which vertex is isolated? Slide 149 / 192
Slide 150 / 192 Properties Return to Table of Contents 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. Slide 151 / 192 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. Complete Graph The number of edges of a complete graph is Slide 152 / 192
66 The Duggers, who are huggers, had a family reunion. 50 family members attended. How many hugs were exchanged? Slide 153 / 192 Degrees The degree of a vertex is the number edges that have the vertex as an endpoint. Slide 154 / 192 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? 67 What is the degree of A? Slide 155 / 192 A C B
68 What is the degree of B? Slide 156 / 192 A C B 69 What is the degree of C? Slide 157 / 192 A C B 70 What is the degree of the network? Slide 158 / 192 A C B
Slide 159 / 192 Corollaries: the degree of a network is even a network will have an even number of odd vertices Can odd number of people at a party shake hands with an odd number of people? Slide 160 / 192 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? Slide 161 / 192 Euler Return to Table of Contents
Konisberg Bridge Problem Slide 162 / 192 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? In 1736, 19 year old Leonhard Euler, one of the greatest mathematicians of all time, solve the problem. Slide 163 / 192 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. Traversable A network is traversable if each edge can be traveled travelled exactly once. Slide 164 / 192 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.
Euler determined that it was not possible because there are 4 odd vertices. Slide 165 / 192 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. Slide 167 / 192 For a network to be an Euler circuit, every vertex has an even degree.
71 Which is a walk from v 1 to v 5? Slide 168 / 192 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 72 Is this graph traversable? Slide 169 / 192 Yes No v 3 v 1 e 4 e 3 e 1 e5 v 4 v 2 e 7 v 5 e 8 Slide 170 / 192 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 Connected graphs have all connected vertices v 5 e 8
For all Polyhedra, Euler's Formula Slide 171 / 192 V - E + F = 2 V is the number vertices E is the number of edges F is the number of faces Pentagonal Prism Tetrahedron 10-15 + 7 = 2 4-6 + 4 =2 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 Slide 172 / 192 V=7 E=9 F=3+1 73 How many 'faces' does this graph have? Slide 173 / 192
74 How many 'edges' does this graph have? Slide 174 / 192 75 How many 'vertices' does this graph have? Slide 175 / 192 76 For this graph, what does V - E + F=? Slide 176 / 192
Slide 177 / 192 Matrix Powers and Walks Return to Table of Contents Slide 178 / 192 There are also adjacency matrices for undirected graphs. Slide 179 / 192 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?
The number of walks of length 1 from a 1 to a 3 is 3. a 1 Slide 180 / 192 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 3. 77 How many walks of length 2 are there from a 2 to a 4? Slide 181 / 192 a 1 a 2 a 4 a 3 78 How many walks of length 3 are there from a 2 to a 2? Slide 182 / 192 a 1 a 2 a 4 a 3
79 How many walks of length 5 are there from a 1 to a 3? Slide 183 / 192 a 1 a 2 a 4 a 3 Slide 184 / 192 Markov Chains Return to Table of Contents 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..40.10 < C <.40 < <.30.60 <.40 < <.10 I <.20 Slide 185 / 192 < O.50 <
What is the probability that a car commercial follows an Internet commercial? Slide 186 / 192.10 < C <.40 <.40 <.30.60 <.40 < <.10 I <.20 < O.50 < Slide 187 / 192.10 < What will the commercial be 2 commercials after a car ad? Using the properties from walks, square the transition matrix..40 C <.40 < <.30.60 <.40 < <.10 I <.20 Slide 188 / 192 < O.50 The first row gives the likelihood of the type of ad following a car ad. <
This method can be applied for any number of ads. But notice what happens to the elements as we get to 10 ads away. Slide 189 / 192 This means that no matter what commercial is on, there is an 18% chance that 10 ads from now will be an Internet ad. < 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. Slide 190 / 192 Make a graph and a transition matrix. 80 Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born in 10 generations? Slide 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? Slide 192 / 192