Pre-Calculus - Center For Teaching & Learningcontent.njctl.org/courses/math/pre-calculus/matrices/matrices-3/matrices-3-2015-03-24...Pre-Calculus Matrices 2015-03-23 Slide 3 / 192

Slide 1 / 192 Slide 2 / 192

Pre-Calculus

Matrices

www.njctl.org

2015-03-23

Slide 3 / 192

Table of Content

Introduction to MatricesMatrix Arithmetic

AdditionSubtraction

Scalar Multiplication

MultiplicationSolving Systems of Equations using Matrices

Finding Determinants of 2x2 & 3x3Finding the Inverse of 2x2 & 3x3Representing 2- and 3-variable systemsSolving Matrix Equations

Circuits

Slide 4 / 192

CircuitsTable of Content

DefinitionPropertiesEulerMatrix Powers and WalksMarkov Chains

Slide 5 / 192

Introduction to Matrices

Return to Table of Contents

Slide 6 / 192

A matrix is an ordered array.

The matrix consists of rows and columns.

Rows

Columns

This matrix has 3 rows and 3 columns, it is said to be 3x3.

http://www.njctl.org




page3svg

page4svg

page6svg

page7svg

page5svg

page8svg

page9svg

page10svg

page11svg

page12svg

page13svg

page2svg

page15svg

page17svg

page18svg

page19svg

page20svg

page127svg

page1svg

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


Slide 11 / 192


Slide 12 / 192


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 Slide 18 / 192


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 m3,4

Slide 21 / 192 Slide 22 / 192

11 Identify the number in the given position.

Slide 23 / 192


Slide 24 / 192


Slide 25 / 192


Slide 26 / 192

Matrix Arithmetic


Slide 27 / 192

Scalar Multiplication


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.

Slide 29 / 192

Try These

Slide 30 / 192

Given: find 6A

Let B = 6A, find b1,2

page1svg

page1svg

Slide 31 / 192

15 Find the given element.

Slide 32 / 192


Slide 33 / 192


Slide 34 / 192


Slide 35 / 192

Addition


Slide 36 / 192

page1svg

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


Slide 41 / 192


Slide 42 / 192


Slide 43 / 192

Subtraction


Slide 44 / 192To 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.

Slide 45 / 192 Slide 46 / 192

23 Subtract the following matrices and find the given element.

Slide 47 / 192


Slide 48 / 192


page1svg

Slide 49 / 192


Slide 50 / 192

Slide 51 / 192

27 Perform the following operations on the given matrices and find the given element.

Slide 52 / 192


Slide 53 / 192


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?

A yes, 3x3

B yes, 4x4

C yes, 3x4

D they cannot be multiplied

Slide 59 / 192


A yes, 3x3

B yes, 4x4

C yes, 3x4


Slide 60 / 192


A yes, 3x3

B yes, 4x4

C yes, 3x4


page1svg

Slide 61 / 192


A yes, 3x3

B yes, 4x4

C yes, 3x4


Slide 62 / 192

To multiply matrices, distribute the rows of first to the columns of the second.

Add the products.

Slide 63 / 192

Try These

Slide 64 / 192

Try These

Slide 65 / 192 Slide 66 / 192


Slide 67 / 192


Slide 68 / 192


Slide 69 / 192


Slide 70 / 192

Solving Systems of Equations

using MatricesReturn to Table of Contents

Slide 71 / 192

Finding Determinants of

2x2 & 3x3Return 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.

page1svg

page1svg

Slide 73 / 192 Slide 74 / 192

Try These:

Slide 75 / 192

39 Find the determinant of the following:

Slide 76 / 192


Slide 77 / 192


Slide 78 / 192


Slide 79 / 192 Slide 80 / 192

Slide 81 / 192Eliminate 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 / 192Begin 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 / 192Begin 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


Slide 87 / 192


Slide 88 / 192


Slide 89 / 192


Slide 90 / 192

Finding the Inverse of 2x2 & 3x3


page1svg

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:



Slide 92 / 192Property of the IdentityMatrix

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

Slide 97 / 192

Find the inverse of matrix A

Slide 98 / 192


Slide 99 / 192


Slide 100 / 192


Slide 101 / 192

Inverse of a 3x3 MatrixThis 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

Inverse of a 3x3 MatrixThink 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 103 / 192 Slide 104 / 192

Slide 105 / 192

We began with this:

We ended with this:

Meaning the inverse of is

Slide 106 / 192

Find the inverse of:

Slide 107 / 192

Find the inverse of:

Slide 108 / 192

Representing 2- and 3-Variable Systems


page1svg

Slide 109 / 192

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.

page1svg

Slide 115 / 192Find x and y

Slide 116 / 192Find x and y

Slide 117 / 192

47 Is this system ready to be made into a matrix equation?

Yes

No

Slide 118 / 192

48 Which of the following is the correct matrix equation for the system?

A

B

C

D

Slide 119 / 192

49 What is the determinant of:

A -17

B -13

C 13

D 17

Slide 120 / 192

50 What is the inverse of:

A

B

C

D

Slide 121 / 192

51 Find the solution to What is the x-value?

Slide 122 / 192

52 Find the solution to What is the y-value?

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

B

C

D

Slide 125 / 192

55 What is the determinant of:

A -10

B -2

C 2

D 10

Slide 126 / 192

56 What is the inverse of:

A

B

C

D

Slide 127 / 192

57 Find the solution to What is the x-value?

Slide 128 / 192

58 Find the solution to What is the y-value?

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

Start

Swap Rows 1&2

Subtract 5 times row 1 from row 2

Subtract row 1 from row 2

Swapped row 2 and 3

(rather divide by 3 than 7)

Divide row 2 by -3

Add 7 times row 2 to row 3

Subtract 2 times row 2 from row 1

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.

Slide 132 / 192

Convert the system to an augmented matrice. Solve using row reduction

Slide 133 / 192


Slide 134 / 192


Slide 135 / 192

Circuits


Slide 136 / 192

Definition


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.

Slide 138 / 192

page1svg

page1svg

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.

Slide 140 / 192

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

Slide 141 / 192

59 Which edge(s) are loops?

A e1

B e2

C e3

D e4

E e5

F e6

G v1

H v2

I v3

J v4

Slide 142 / 192

60 Which edge(s) are parallel?

A e1

B e2

C e3

D e4

E e5

F e6

G v1

H v2

I v3

J v4

Slide 143 / 192

61 Which edge(s) are adjacent to e4?

A e1

B e2

C e3

D e4

E e5

F e6

G v1

H v2

I v3

J v4

Slide 144 / 192

62 Which vertices are adjacent to v4?

A e1

B e2

C e3

D e4

E e5

F e6

G v1

H v2

I v3

J v4

Slide 145 / 192

63 Which vertex is isolated?

A e1

B e2

C e3

D e4

E e5

F none

G v1

H v2

I v3

J v4

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.

Slide 147 / 192 Slide 148 / 192

64 How many paths are there from v2 to v3?

Slide 149 / 192

65 Which vertex is isolated?

Slide 150 / 192

Properties


page1svg

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.

Slide 152 / 192

Complete GraphThe 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

The degree of a vertex is the number edges that have the vertex as an endpoint.

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?

Degrees

Loops count as 2.

Slide 155 / 192

67 What is the degree of A?

A

B

C

Slide 156 / 192

68 What is the degree of B?

A

B

C

Slide 157 / 192

69 What is the degree of C?

A

B

C

Slide 158 / 192

70 What is the degree of the network?

A

B

C

Slide 159 / 192

Corollaries:· the degree of a network is even· a network will have an even number of odd vertices

Slide 160 / 192Can odd number of people at a party shake hands with an odd number of people?

Corollaries:· the degree of a network is even· a network will have an even number of odd vertices

Think about the corollaries.

An odd number of people means how many vertices?

An odd number of handshakesmeans what is the degreeof those verticces?

Slide 161 / 192

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.

The Konisberg Bridge Problem asks if it is possible to travel each bridge exactly once and end up back where you started?

page1svg

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.

Slide 164 / 192

TraversableA 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.

Slide 165 / 192

Euler determined that it was not possible because there are 4 odd vertices.

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.

Slide 168 / 192

v1

v2v4

v5

v3

e4

e3

e1e5

e7

e8

71 Which is a walk from v1 to v5?

A v1,e3,v3,e4,v5

B v1,e2,v2,e3,v3,e5,v4,e7,v5

C v1,e3,e2,e7,v5

D v1,e3,v3,e5,v4,e7,v5

e2

Slide 169 / 192

72 Is this graph traversable?

Yes

Nov1

v2v4

v5

v3

e4

e3

e1e5

e7

e8

Slide 170 / 192

Connected vertices have at least on walk connecting them.

Connected graphs have all connected vertices

v1

v2v4

v5

v3

e4

e3

e1e5

e7

e8

Slide 171 / 192

Euler's Formula

V - E + F = 2V is the number verticesE is the number of edgesF is the number of faces

For all Polyhedra,

TetrahedronPentagonal Prism

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.

Euler's Formula

V - E + F = 2V is the number verticesE is the number of edgesF is the number of faces

V=5E=7F=3+1

V=7E=9F=3+1

Slide 173 / 192

73 How many 'faces' does this graph have?

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

76 For this graph, what does V - E + F= ?

Slide 177 / 192

Matrix Powers and Walks


Slide 178 / 192

Slide 179 / 192

There are also adjacency matrices for undirected graphs.

a1

a2

a3

a4

maindiagonal

What do the numbers on the main diagonal represent?

What can be said about the halves of adjacency matrix?

Slide 180 / 192The number of walks of length 1 from a1 to a3 is 3. a1

a3

a4a2

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.

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 a3.

page1svg

Slide 181 / 192

77 How many walks of length 2 are there from a2 to a4?

a1

a3

a4a2

Slide 182 / 192


a1

a3

a4a2

Slide 183 / 192


a1

a3

a4a2

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.

<

< <

<

< <

< <

<

C I

O

.40

.30.20

.40

.10

.50

.60.10

.40

<

Slide 186 / 192

<

< <

<

< <

< <

<

C I

O

.40

.30.20

.40

.10

.50

.60.10

.40

<

What is the probability that a car commercial follows an Internet commercial?

page1svg

Slide 187 / 192 Slide 188 / 192

<

< <

<

< <

< <

<

C I

O

.40

.30.20

.40

.10

.50

.60.10

.40

<

What will the commercial be 2 commercials after a car ad? Using the properties from walks, square the transition matrix.

The first row gives the likelihood of the type of ad following a car ad.

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.

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?

Documents

Pre-Calculus - Center For Teaching & Learningcontent.njctl.org/courses/math/pre-calculus/matrices/matrices-3/matrices-3-2015-03-24...Pre-Calculus Matrices 2015-03-23 Slide 3 / 192