Post on 17-May-2019
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PreCalculus
Matrices
www.njctl.org
20150323
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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 3variable systemsSolving Matrix Equations
Circuits
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CircuitsTable of Content
DefinitionPropertiesEulerMatrix Powers and WalksMarkov Chains
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Introduction to Matrices
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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.
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What are the dimensions of the following matrices?
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1 How many rows does the following matrix have?
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2 How many columns does the following matrix have?
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3 How many rows does the following matrix have?
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4 How many columns does the following matrix have?
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5 How many rows does the following matrix have?
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6 How many columns does the following matrix have?
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Matrices can be named with a capital letter.
A subscript can be used to tell the dimensions of the matrix
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How many rows does each matrix have? How many columns?
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7 How many rows does the following matrix have?
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8 How many columns does the following matrix have?
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9 How many rows does the following matrix have?
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10 How many columns does the following matrix have?
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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
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11 Identify the number in the given position.
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12 Identify the number in the given position.
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13 Identify the number in the given position.
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14 Identify the number in the given position.
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Matrix Arithmetic
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Scalar Multiplication
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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.
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Try These
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Given: find 6A
Let B = 6A, find b1,2
Answer
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15 Find the given element.
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16 Find the given element.
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17 Find the given element.
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18 Find the given element.
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Addition
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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.
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After checking to see addition is possible,add the corresponding elements.
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19 Add the following matrices and find the given element.
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20 Add the following matrices and find the given element.
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21 Add the following matrices and find the given element.
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22 Add the following matrices and find the given element.
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Subtraction
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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.
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23 Subtract the following matrices and find the given element.
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24 Subtract the following matrices and find the given element.
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25 Subtract the following matrices and find the given element.
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26 Subtract the following matrices and find the given element.
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27 Perform the following operations on the given matrices and find the given element.
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28 Perform the following operations on the given matrices and find the given element.
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29 Perform the following operations on the given matrices and find the given element.
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30 Perform the following operations on the given matrices and find the given element.
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Multiplication
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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.
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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?
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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
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32 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
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33 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
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34 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
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To multiply matrices, distribute the rows of first to the columns of the second.
Add the products.
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Try These
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Try These
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Try These
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35 Perform the following operations on the given matrices and find the given element.
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36 Perform the following operations on the given matrices and find the given element.
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37 Perform the following operations on the given matrices and find the given element.
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38 Perform the following operations on the given matrices and find the given element.
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Solving Systems of Equations
using MatricesReturn to Table of Contents
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Finding Determinants of
2x2 & 3x3Return to Table of Contents
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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.
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To find the determinant of a 2x2 matrix:The product of the primary diagonal minus the product of the secondary diagonal.
Example:
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Try These:
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39 Find the determinant of the following:
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40 Find the determinant of the following:
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41 Find the determinant of the following:
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42 Find the determinant of the following:
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Finding the Determinant of a 3x3 MatrixUse 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.
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Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants.Solve.
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Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants.Solve.
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Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants.Solve.
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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.
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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.
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43 Find the determinant of the following:
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44 Find the determinant of the following:
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45 Find the determinant of the following:
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46 Find the determinant of the following:
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Finding the Inverse of 2x2 & 3x3
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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:
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Property of the IdentityMatrix
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The inverse of matrix A is matrix A1.The product of a matrix and its inverse is the identity matrix, I.
example:
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Note: Not all matrices have an inverse.• matrix must be square• the determinant of the matrix cannot = 0
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Finding the inverse of a 2x2 matrix
Example: Find the inverse of matrix M.
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check:
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Find the inverse of matrix A
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Find the inverse of matrix A
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Find the inverse of matrix A
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Find the inverse of matrix A
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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.
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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 nonzero 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.
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Now we know the rules, let's play.
Beginning matrix
Switched rows 1&2
Divided row 1 by 4
Subtracted 2 times row 1 from row 2
Subtracted 6 times row 1 from row 3
Switched rows 2&3
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Cont. from
previous slide
Divrow 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 1.1875 times row 3 from row 1
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We began with this:
We ended with this:
Meaning the inverse of is
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Find the inverse of:
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Find the inverse of:
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Representing 2 and 3Variable Systems
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Solving Matrix Equations
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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 constantsvariables
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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:
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Solve:
Step 1:
Step 2: find the inverse of
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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.
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Rewrite each system as a product of matrices.
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Find x and y
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Find x and y
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47 Is this system ready to be made into a matrix equation?
Yes
No
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48 Which of the following is the correct matrix equation for the system?
A
B
C
D
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49 What is the determinant of:
A 17
B 13
C 13
D 17
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50 What is the inverse of:
A
B
C
D
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51 Find the solution to What is the xvalue?
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52 Find the solution to What is the yvalue?
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53 Is this system ready to be made into a matrix equation?
Yes
No
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54 Which of the following is the correct matrix equation for the system?
A
B
C
D
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55 What is the determinant of:
A 10
B 2
C 2
D 10
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56 What is the inverse of:
A
B
C
D
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57 Find the solution to What is the xvalue?
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58 Find the solution to What is the yvalue?
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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.
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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
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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.
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Convert the system to an augmented matrice. Solve using row reduction
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Convert the system to an augmented matrice. Solve using row reduction
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Convert the system to an augmented matrice. Solve using row reduction
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Circuits
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Definition
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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.
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Edge endpoints
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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.
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Make a simple graph with vertices {a, b, c, d} and as many edges as possible.
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59 Which edge(s) are loops?
A e1B e2C e3D e4E e5F e6
G v1H v2I v3J v4
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60 Which edge(s) are parallel?
A e1B e2C e3D e4E e5F e6
G v1H v2I v3J v4
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61 Which edge(s) are adjacent to e4?
A e1B e2C e3D e4E e5F e6
G v1H v2I v3J v4
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62 Which vertices are adjacent to v4?
A e1B e2C e3D e4E e5F e6
G v1H v2I v3J v4
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63 Which vertex is isolated?
A e1B e2C e3D e4E e5F none
G v1H v2I v3J v4
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Some graphs will show that an edge can be traversed in only one direction, like one way streets.
This is a directed graph.
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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.
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64 How many paths are there from v2 to v3?
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65 Which vertex is isolated?
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Properties
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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.
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Complete GraphThe number of edges of a complete graph is
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66 The Duggers, who are huggers, had a family reunion. 50 family members attended. How many hugs were exchanged?
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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.
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67 What is the degree of A?
A
B
C
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68 What is the degree of B?
A
B
C
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69 What is the degree of C?
A
B
C
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70 What is the degree of the network?
A
B
C
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Corollaries:• the degree of a network is even• a network will have an even number of odd vertices
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Can 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?
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Euler
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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?
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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.
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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.
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Euler determined that it was not possible because there are 4 odd vertices.
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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.
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For a network to be an Euler circuit, every vertex has an even degree.
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v1
v2v4
v5
v3e4
e3
e1e5
e7
e8
71 Which is a walk from v1 to v5?
A v1,e3,v3,e4,v5B v1,e2,v2,e3,v3,e5,v4,e7,v5C v1,e3,e2,e7,v5D v1,e3,v3,e5,v4,e7,v5
e2
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72 Is this graph traversable?
Yes
No v1
v2v4
v5
v3e4
e3
e1e5
e7
e8
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Connected vertices have at least on walk connecting them.
Connected graphs have all connected vertices
v1
v2v4
v5
v3e4
e3
e1e5
e7
e8
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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
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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
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73 How many 'faces' does this graph have?
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74 How many 'edges' does this graph have?
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75 How many 'vertices' does this graph have?
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76 For this graph, what does V E + F= ?
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Matrix Powers and Walks
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Earlier in this unit, we looked at adjacency matrices for directed graphs.
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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?
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The number of walks of length 1 from a1 to a3 is 3. a1
a3
a4a2How 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.
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77 How many walks of length 2 are there from a2 to a4?
a1
a3
a4a2
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78 How many walks of length 3 are there from a2 to a2?
a1
a3
a4a2
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79 How many walks of length 5 are there from a1 to a3?
a1
a3
a4a2
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Markov Chains
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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.
<
< <
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C I
O
.40
.30.20
.40
.10
.50
.60.10
.40
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< <
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C I
O
.40
.30.20
.40
.10
.50
.60.10
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What is the probability that a car commercial follows an Internet commercial?
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< <
< <
<
C I
O
.40
.30.20
.40
.10
.50
.60.10
.40
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An adjacency matrix that shows the probabilities of what happens next is called a transition matrix.
Since each row adds to 1(100%) this matrix is also called a stochastic matrix.
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< <
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C I
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.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.
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<
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.
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Horse breeders found that the if a champion horse sired an offspring it had 40% of being a champion. If a nonchampion horses had offspring, they were 35% likely of being champions.
Make a graph and a transition matrix.
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80 Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born in 10 generations?
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81 Using the transition matrix for champion horse breeding, what is the likelihood of a champion being born to nonchampions in 2 generations?