1
PreCalculus Matrices
20150323 www.njctl.org
2
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 3variable systems Solving Matrix Equations Circuits
3
Table of Content
Circuits Definition Properties Euler Matrix Powers and Walks Markov Chains
4
Introduction to Matrices Return to Table of Contents
5
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.
6
What are the dimensions of the following matrices?
7
1
How many rows does the following matrix have?
8
2
How many columns does the following matrix have?
9
3
How many rows does the following matrix have?
10
4
How many columns does the following matrix have?
11
5
How many rows does the following matrix have?
12
6
How many columns does the following matrix have?
13
Matrices can be named with a capital letter.
A subscript can be used to tell the dimensions of the matrix
14
How many rows does each matrix have? How many columns?
15
7
How many rows does the following matrix have?
16
8
How many columns does the following matrix have?
17
9
How many rows does the following matrix have?
18
10
How many columns does the following matrix have?
19
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
20
21
11
Identify the number in the given position.
22
12
Identify the number in the given position.
23
13
Identify the number in the given position.
24
14
Identify the number in the given position.
25
Matrix Arithmetic Return to Table of Contents
26
Scalar Multiplication Return to Table of Contents
27
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.
28
Try These
29
find 6A
Answer
Given:
Let B = 6A, find b 1,2
30
15
Find the given element.
31
16
Find the given element.
32
17
Find the given element.
33
18
Find the given element.
34
Addition Return to Table of Contents
35
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.
36
After checking to see addition is possible, add the corresponding elements.
37
38
19
Add the following matrices and find the given element.
39
20
Add the following matrices and find the given element.
40
21
Add the following matrices and find the given element.
41
22
Add the following matrices and find the given element.
42
Subtraction Return to Table of Contents
43
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.
44
45
23
Subtract the following matrices and find the given element.
46
24
Subtract the following matrices and find the given element.
47
25
Subtract the following matrices and find the given element.
48
26
Subtract the following matrices and find the given element.
49
50
27
Perform the following operations on the given matrices and find the given element.
51
28
Perform the following operations on the given matrices and find the given element.
52
29
Perform the following operations on the given matrices and find the given element.
53
30
Perform the following operations on the given matrices and find the given element.
54
Multiplication Return to Table of Contents
55
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.
56
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?
57
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
58
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
59
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
60
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
61
To multiply matrices, distribute the rows of first to the columns of the second. Add the products.
62
Try These
63
Try These
64
Try These
65
35
Perform the following operations on the given matrices and find the given element.
66
36
Perform the following operations on the given matrices and find the given element.
67
37
Perform the following operations on the given matrices and find the given element.
68
38
Perform the following operations on the given matrices and find the given element.
69
Solving Systems of Equations using Matrices Return to Table of Contents
70
Finding Determinants of 2x2 & 3x3 Return to Table of Contents
71
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.
72
To find the determinant of a 2x2 matrix: The product of the primary diagonal minus the product of the secondary diagonal.
Example:
73
Try These:
74
39
Find the determinant of the following:
75
40
Find the determinant of the following:
76
41
Find the determinant of the following:
77
42
Find the determinant of the following:
78
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.
79
80
Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve.
81
Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve.
82
Eliminate the appropriate row and column in each. Rewrite as 3 2x2 determinants. Solve.
83
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.
84
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
43
Find the determinant of the following:
86
44
Find the determinant of the following:
87
45
Find the determinant of the following:
88
46
Find the determinant of the following:
89
Finding the Inverse of 2x2 & 3x3 Return to Table of Contents
90
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:
91
Property of the IdentityMatrix
92
The inverse of matrix A is matrix A1. The product of a matrix and its inverse is the identity matrix, I.
example:
93
Note: Not all matrices have an inverse. • matrix must be square • the determinant of the matrix cannot = 0
94
Finding the inverse of a 2x2 matrix
Example: Find the inverse of matrix M.
95
check:
96
Find the inverse of matrix A
97
Find the inverse of matrix A
98
Find the inverse of matrix A
99
Find the inverse of matrix A
100
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.
101
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 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.
102
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
103
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 .625 times row 3 from row 2 Sub 1.1875 times row 3 from row 1
104
We began with this:
We ended with this:
Meaning the inverse of is
105
Find the inverse of:
106
Find the inverse of:
107
Representing 2 and 3 Variable Systems Return to Table of Contents
108
Solving Matrix Equations Return to Table of Contents
109
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
110
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:
111
Solve:
Step 1: Step 2: find the inverse of
112
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.
113
Rewrite each system as a product of matrices.
114
Find x and y
115
Find x and y
116
47
Is this system ready to be made into a matrix equation?
Yes No
117
48
Which of the following is the correct matrix equation for the system?
A
C
B
D
118
49
What is the determinant of: A
17
B
13
C
13
D
17
119
50
What is the inverse of: A
B
C
D
120
51
Find the solution to What is the xvalue?
121
52
Find the solution to What is the yvalue?
122
53
Is this system ready to be made into a matrix equation?
Yes No
123
54
Which of the following is the correct matrix equation for the system?
A
C
B
D
124
55
What is the determinant of: A
10
B
2
C
2
D
10
125
56
What is the inverse of: A
B
C
D
126
57
Find the solution to What is the xvalue?
127
58
Find the solution to What is the yvalue?
128
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.
129
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
130
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.
131
Convert the system to an augmented matrice. Solve using row reduction
132
Convert the system to an augmented matrice. Solve using row reduction
133
Convert the system to an augmented matrice. Solve using row reduction
134
Circuits Return to Table of Contents
135
Definition Return to Table of Contents
136
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.
137
Edge
endpoints
138
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.
139
Make a simple graph with vertices {a, b, c, d} and as many edges as possible.
140
59
Which edge(s) are loops? A e1
G
v1
B e2
H
v2
C e3
I v3
D e4
J
v4
E e5
F
e6
141
60
Which edge(s) are parallel? A e1
G
v1
B e2
H
v2
C e3
I v3
D e4
J
v4
E e5
F
e6
142
61
Which edge(s) are adjacent to e4? A e1
G
v1
B e2
H
v2
C e3
I v3
D e4
J
v4
E e5
F
e6
143
62
Which vertices are adjacent to v4? A e1
G
v1
B e2
H
v2
C e3
I v3
D e4
J
v4
E e5
F
e6
144
63
Which vertex is isolated? A e1
G
v1
B e2
H
v2
C e3
I v3
D e4
J
v4
E e5
F
none
145
Some graphs will show that an edge can be traversed in only one direction, like one way streets.
This is a directed graph.
146
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.
147
64
How many paths are there from v2 to v3?
148
65
Which vertex is isolated?
149
Properties Return to Table of Contents
150
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.
151
Complete Graph The number of edges of a complete graph is
152
66
The Duggers, who are huggers, had a family reunion. 50 family members attended. How many hugs were exchanged?
153
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?
154
67
What is the degree of A?
A C
B
155
68
What is the degree of B?
A C
B
156
69
What is the degree of C?
A C
B
157
70
What is the degree of the network?
A C
B
158
Corollaries: • the degree of a network is even • a network will have an even number of odd vertices
159
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?
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?
160
Euler Return to Table of Contents
161
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?
162
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.
163
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.
164
Euler determined that it was not possible because there are 4 odd vertices.
165
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.
166
For a network to be an Euler circuit, every vertex has an even degree.
167
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
v1
e2
e4
C v1,e3,e2,e7,v5 D v1, e3,v3,e5,v4,e7,v5
e3
v3
e5
e1
v4
v2
e7 v5
e8
168
72
Is this graph traversable?
Yes No
e3
v3
v1
e4 e5
e1
v4
v2
e7 v5
e8
169
Connected vertices have at least on walk connecting them.
e3
v3
v1
e4 e5
e1
v4
v2
e7 v5
e8
Connected graphs have all connected vertices
170
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
10 15 + 7 = 2
Tetrahedron
4 6 + 4 =2
171
Apply Euler's Formula to circuits. Add 1 to faces for the not enclosed region.
Euler's Formula V E + F = 2 V is the number vertices E is the number of edges F is the number of faces
V=5 E=7 F=3+1
V=7 E=9 F=3+1
172
73
How many 'faces' does this graph have?
173
74
How many 'edges' does this graph have?
174
75
How many 'vertices' does this graph have?
175
76
For this graph, what does V E + F= ?
176
Matrix Powers and Walks Return to Table of Contents
177
Earlier in this unit, we looked at adjacency matrices for directed graphs.
178
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?
179
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 a3.
180
77
How many walks of length 2 are there from a2 to a4?
a1 a4
a2 a3
181
78
How many walks of length 3 are there from a2 to a2?
a1 a4
a2 a3
182
79
How many walks of length 5 are there from a1 to a3?
a1 a4
a2 a3
183
Markov Chains Return to Table of Contents
184
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
C
.60
.20
![[TOPICS IN PRE-CALCULUS] Functions, Graphs, and Basic Mensuration Formulas](https://kipdf.com/img/300x300/topics-in-pre-calculus-functions-graphs-and-basic-_5add3db77f8b9ab1588b45f5.jpg)
