Slide 1 / 192 Slide 2 / 192 Pre-Calculus Matrices 2015-03-23 www.njctl.org Slide 3 / 192 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 4 / 192 Content Circuits Definition Properties Euler Matrix Powers and Walks Markov Chains Slide 5 / 192 Slide 6 / 192 A matrix is an ordered array. The matrix consists of rows and columns. Columns Introduction to Matrices Rows This matrix has 3 rows and 3 columns, it is said to be 3x3.
Slide 7 / 192 What are the dimensions of the following matrices? Slide 8 / 192 1 How many rows does the following matrix have? Slide 9 / 192 2 How many columns does the following matrix have? Slide 10 / 192 3 How many rows does the following matrix have? Slide 11 / 192 4 How many columns does the following matrix have? Slide 12 / 192 5 How many rows does the following matrix have?
Slide 13 / 192 Slide 14 / 192 6 How many columns does the following matrix have? Slide 15 / 192 Slide 16 / 192 How many rows does each matrix have? How many columns? Slide 17 / 192 Slide 18 / 192 9 How many rows does the following matrix have?
Slide 19 / 192 10 How many columns does the following matrix have? 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 Slide 22 / 192 11 Identify the number in the given position. Slide 23 / 192 12 Identify the number in the given position. Slide 24 / 192 13 Identify the number in the given position.
Slide 25 / 192 Slide 26 / 192 14 Identify the number in the given position. Matrix Arithmetic Slide 27 / 192 Slide 28 / 192 A scalar multiple is when a single number is multiplied to the entire matrix. Scalar Multiplication To multiply by a scalar, distribute the number to each entry in the matrix. Slide 29 / 192 Slide 30 / 192 Try These Given: find 6A Let B = 6A, find b 1,2
Slide 30 (Answer) / 192 Slide 31 / 192 Given: find 6A 15 Find the given element. Answer Let B = 6A, find b 1,2-6 [This object is a pull tab] 16 Find the given element. Slide 32 / 192 17 Find the given element. Slide 33 / 192 Slide 34 / 192 Slide 35 / 192 18 Find the given element. Addition
Slide 36 / 192 Slide 37 / 192 After checking to see addition is possible, add the corresponding elements. Slide 38 / 192 Slide 39 / 192 19 Add the following matrices and find the given element. Slide 40 / 192 20 Add the following matrices and find the given element. Slide 41 / 192 21 Add the following matrices and find the given element.
Slide 42 / 192 Slide 43 / 192 22 Add the following matrices and find the given element. Subtraction Slide 44 / 192 To be able to subtract matrices, they must have the same dimensions, like addition. Slide 45 / 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 46 / 192 23 Subtract the following matrices and find the given element. Slide 47 / 192 24 Subtract the following matrices and find the given element.
Slide 48 / 192 25 Subtract the following matrices and find the given element. Slide 49 / 192 26 Subtract the following matrices and find the given element. Slide 50 / 192 Slide 51 / 192 27 Perform the following operations on the given matrices and find the given element. Slide 52 / 192 28 Perform the following operations on the given matrices and find the given element. Slide 53 / 192 29 Perform the following operations on the given matrices and find the given element.
Slide 54 / 192 Slide 55 / 192 Multiplication Slide 56 / 192 Slide 57 / 192 Slide 58 / 192 31 Can the given matrices be multiplied and if so,what size will the matrix of their product be? Slide 59 / 192 32 Can the given matrices be multiplied and if so,what size will the matrix of their product be? A yes, 3x3 A yes, 3x3 B yes, 4x4 B yes, 4x4 C yes, 3x4 C yes, 3x4 D they cannot be multiplied D they cannot be multiplied
Slide 60 / 192 33 Can the given matrices be multiplied and if so,what size will the matrix of their product be? Slide 61 / 192 34 Can the given matrices be multiplied and if so,what size will the matrix of their product be? A yes, 3x3 A yes, 3x3 B yes, 4x4 B yes, 4x4 C yes, 3x4 C yes, 3x4 D they cannot be multiplied D they cannot be multiplied Slide 62 / 192 Slide 63 / 192 To multiply matrices, distribute the rows of first to the columns of the second. Add the products. Try These Slide 64 / 192 Slide 65 / 192 Try These
Slide 66 / 192 35 Perform the following operations on the given matrices and find the given element. Slide 67 / 192 36 Perform the following operations on the given matrices and find the given element. Slide 68 / 192 37 Perform the following operations on the given matrices and find the given element. Slide 69 / 192 38 Perform the following operations on the given matrices and find the given element. Slide 70 / 192 Slide 71 / 192 Solving Systems of Equations using Matrices Finding Determinants of 2x2 & 3x3
Slide 72 / 192 Slide 73 / 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 74 / 192 Slide 75 / 192 Try These: 39 Find the determinant of the following: Slide 76 / 192 40 Find the determinant of the following: Slide 77 / 192 41 Find the determinant of the following:
Slide 78 / 192 Slide 79 / 192 42 Find the determinant of the following: Slide 80 / 192 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 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 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 86 / 192 43 Find the determinant of the following: Slide 87 / 192 44 Find the determinant of the following: Slide 88 / 192 45 Find the determinant of the following: Slide 89 / 192 46 Find the determinant of the following:
Slide 90 / 192 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. Finding the Inverse of 2x2 & 3x3 2x2 Identity Matrix: 3x3 Identity Matrix: 4x4 Identity Matrix: Slide 92 / 192 Property of the IdentityMatrix Slide 93 / 192 Slide 94 / 192 Slide 95 / 192 Note: Not all matrices have an inverse. matrix must be square the determinant of the matrix cannot = 0
Slide 96 / 192 Slide 97 / 192 Find the inverse of matrix A 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 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.
Slide 102 / 192 Slide 103 / 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. Slide 104 / 192 Slide 105 / 192 We began with this: We ended with this: Meaning the inverse of is Slide 106 / 192 Slide 107 / 192 Find the inverse of: Find the inverse of:
Slide 108 / 192 Slide 109 / 192 Representing 2- and 3- Variable Systems Solving Matrix Equations Slide 110 / 192 Slide 111 / 192 Slide 112 / 192 Slide 113 / 192
Slide 114 / 192 Rewrite each system as a product of matrices. Find x and y Slide 115 / 192 Find x and y Slide 116 / 192 Slide 117 / 192 47 Is this system ready to be made into a matrix equation? Yes No Slide 118 / 192 Slide 119 / 192 48 Which of the following is the correct matrix equation for the system? A C 49 What is the determinant of: A -17 B -13 C 13 D 17 B D
50 What is the inverse of: A Slide 120 / 192 51 Find the solution to What is the x-value? Slide 121 / 192 B C D 52 Find the solution to What is the y-value? Slide 122 / 192 Slide 123 / 192 53 Is this system ready to be made into a matrix equation? Yes No Slide 124 / 192 54 Which of the following is the correct matrix equation for the system? A C Slide 125 / 192 55 What is the determinant of: A -10 B -2 C 2 D 10 B D
56 What is the inverse of: A Slide 126 / 192 57 Find the solution to What is the x-value? Slide 127 / 192 B C D 58 Find the solution to What is the y-value? Slide 128 / 192 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. Slide 130 / 192 Slide 131 / 192 Start Swapped row 2 and 3 (rather divide by 3 than 7) From Previous slide Swap Rows 1&2 Divide row 2 by -3 Divide row 3 by -37/3 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 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.
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 Convert the system to an augmented matrice. Solve using row reduction Circuits Slide 136 / 192 Slide 137 / 192 A Graph of a network consists of vertices (points) and edges (edges connect the points) Definition The points marked v are the vertices, or nodes, of the network. The edges are e.
Slide 138 / 192 Slide 139 / 192 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. Slide 140 / 192 Make a simple graph with vertices {a, b, c, d} and as many edges as possible. 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 Slide 142 / 192 60 Which edge(s) are parallel? Slide 143 / 192 61 Which edge(s) are adjacent to e 4? A e 1 G v 1 A e 1 G v 1 B e 2 H v 2 B e 2 H v 2 C e 3 I v 3 C e 3 I v 3 D e 4 J v 4 D e 4 J v 4 E e 5 E e 5 F e 6 F e 6
Slide 144 / 192 62 Which vertices are adjacent to v 4? 63 Which vertex is isolated? Slide 145 / 192 A e 1 G v 1 A e 1 G v 1 B e 2 H v 2 B e 2 H v 2 C e 3 I v 3 C e 3 I v 3 D e 4 J v 4 D e 4 J v 4 E e 5 E e 5 F e 6 F none Slide 146 / 192 Slide 147 / 192 Some graphs will show that an edge can be traversed in only one direction, like one way streets. This is a directed graph. Slide 148 / 192 64 How many paths are there from v 2 to v 3? 65 Which vertex is isolated? Slide 149 / 192
Slide 150 / 192 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. Properties 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. Slide 152 / 192 Complete Graph The number of edges of a complete graph is Slide 153 / 192 66 The Duggers, who are huggers, had a family reunion. 50 family members attended. How many hugs were exchanged? Slide 154 / 192 Degrees The degree of a vertex is the number edges that have the vertex as an endpoint. Loops count as 2. 67 What is the degree of A? Slide 155 / 192 A C 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? B
68 What is the degree of B? A Slide 156 / 192 69 What is the degree of C? A Slide 157 / 192 C C B B Slide 158 / 192 Slide 159 / 192 70 What is the degree of the network? A B C Corollaries: the degree of a network is even a network will have an even number of odd vertices Slide 160 / 192 Can odd number of people at a party shake hands with an odd number of people? Slide 161 / 192 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 Euler
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. 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. The Konisberg Bridge Problem asks if it is possible to travel each bridge exactly once and end up back where you started? He then developed rules about traversable graphs. Slide 164 / 192 Traversable A network is traversable if each edge can be traveled travelled exactly once. Slide 165 / 192 Euler determined that it was not possible because there are 4 odd vertices. 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. Slide 166 / 192 Slide 167 / 192 A walk is a sequence of edges and vertices from a to b. A path is a walk with no edge repeated.(traversable) For a network to be an Euler circuit, every vertex has an even degree. 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 168 / 192 71 Which is a walk from v 1 to v 5? 72 Is this graph traversable? Slide 169 / 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 C v 1,e 3,e 2,e 7,v 5 v 3 e 4 e 3 e 1 e5 v 1 e 2 Yes No v 3 e 4 e 3 e 1 e5 v 1 D v 1,e 3,v 3,e 5,v 4,e 7,v 5 v 4 e 7 v 2 v 4 e 7 v 2 v 5 e 8 v 5 e 8 Slide 170 / 192 Connected vertices have at least on walk connecting them. v 4 v 3 e 4 e 3 e 1 e5 e 7 v 1 v 2 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 Tetrahedron Connected graphs have all connected vertices v 5 e 8 10-15 + 7 = 2 4-6 + 4 =2 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 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 173 / 192 73 How many 'faces' does this graph have? V=7 E=9 F=3+1
Slide 174 / 192 74 How many 'edges' does this graph have? Slide 175 / 192 75 How many 'vertices' does this graph have? Slide 176 / 192 Slide 177 / 192 76 For this graph, what does V - E + F=? Matrix Powers and Walks Slide 178 / 192 Slide 179 / 192 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?
Slide 180 / 192 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 Slide 181 / 192 77 How many walks of length 2 are there from a 2 to 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 a 2 a 1 a 4 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. Slide 182 / 192 78 How many walks of length 3 are there from a 2 to a 2? Slide 183 / 192 79 How many walks of length 5 are there from a 1 to a 3? a 1 a 1 a 2 a 4 a 2 a 4 a 3 a 3 Slide 184 / 192 Markov Chains 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..40.10 C.40.30.60 O.40.50.10 I.20
Slide 186 / 192 Slide 187 / 192 What is the probability that a car commercial follows an Internet commercial?.10 C.40.40.30.60.40.10 I.20 O.50 Slide 188 / 192 Slide 189 / 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 O.40.50.10 I.20 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. This means that no matter what commercial is on, there is an 18% chance that 10 ads from now will be an Internet ad. 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. Slide 191 / 192 80 Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born in 10 generations?
Slide 192 / 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?