2.1 - Vertices, Edges, Networks
C7 model real-world situations with networks and matrices (optional)
2.2 - Digraphs, Matrix Set Up and Terminology
B5 develop, analyze, and apply procedures for matrix multiplication (optional)
B6 solve network problems, using matrices (optional)
C7 model real-world situations with networks and matrices (optional)
C17 solve problems using graphing technology
C37 represent network problems using matrices and vice versa (optional)
E6 represent network problems as digraphs (optional)
2.3 - Adding (and Subtracting) Matrices, Scalar multiplication
B5 develop, analyze, and apply procedures for matrix multiplication (optional)
B6 solve network problems, using matrices (optional)
C15 develop and apply strategies for solving problems
Matrices: Terminology
Chapter 2 deals with matrices and how matrix algebra can be used to solve real-life problems. To solve any system of equations, you work with rows and columns. These rows and columns are visible when information is displayed in tables, in graphs or in matrices.
Before performing mathematical operations on matrices, you should highlight the terms that are important to this topic.
· A Matrix is a rectangular array of numbers enclosed by parentheses.
Examples of matrices are:
· The elements or entries of a matrix are the individual numbers in the matrix.
· A row is a horizontal arrangement of the elements in a matrix.
· A column is a vertical arrangement of the elements in a matrix.
· The dimensions of a matrix are the number of rows and the number of columns.
Examples:
The dimensions of A are 2 rows by 3 columns.
This is written as 2 × 3 and read “two by three”.
The dimensions of A are 3 rows by 1 column.
This is written as 3 × 1 and read “three by one”.
Once your students understand these terms, you can introduce them to the mathematical operations.
Network
A network is a set of people, places, objects or ideas that are connected in some way. A network is complete if you can enter and exit it from a single vertex while traveling each edge only once. For example, a road system with crossroads is an example of a network with the roads acting as edges and the intersections, vertices.
Example: A computer software company is having a LAN (local area network) party/event at a community center to promote a new video game. Determine which gamer is unable to play because he is not properly wired to the network.
Solution:
The circled computer is only directly connected
to the computer to his left so will be unable to
compete with the rest of the room.
Exercise:
1. Complete the network
(LAN) by removing one edge and
adding a new edge.
2. A museum has set up series of rooms to showcase displays to celebrate Canadian Native Culture. Is it possible to see all the rooms without going through the same door twice? Why might there be extra doors through which you might not travel?
Digraphs
A directed network is known as a digraph. You can represent the information in a digraph in an adjacency matrix.
Example:
Five students, Andy, Bob, Carol, Doug, andEdith have formed a study group. Andy has Carol and Doug's phone number, Bob has Edith's number, Carol has Doug and Edith's number, Doug has Andy and Bob's number and Edith has everyone's number. Is it possible for Andy get a message to Bob? Explain.
The graph and the adjacency matrix are shown below.
Solution: Andy cannot phone Bob directly but he can call Doug who can call Bob. Andy may also call Carol who can phone Edith and she can call Bob with Andy’s message. A slower way would be for Andy to call Carol to call Doug who can call Bob.
Simplifying a Digraph
Digraphs with many directed edges between adjacent vertices can be simplified. Use numbers to represent the number of edges going in the same direction between two vertices.
Example:
1. Several cargo planes leave Edmonton for cities in Northern Alberta, the Yukon, Northwest Territories and Nunavut. The following flights are made each week:
Edmonton to Fort McMurray (3), Edmonton to Whitehorse, Edmonton to Yellowknife (2), Edmonton to Rankin Inlet, Edmonton to Fort Simpson, Whitehorse to Edmonton, Whitehorse to Yellowknife, Yellowknife to Whitehorse, Yellowknife to Edmonton, Fort McMurray to Yellowknife, Fort Simpson to Whitehorse, Rankin Inlet to Fort McMurray.
Draw a digraph of the information, then simplify.
Solution:
2. Using the digraph from Example 1, answer the following questions:
a. How many trips head north from Edmonton each week?
Solution: 10
b. What is the quickest way to travel from Rankin Inlet to Fort Simpson?
Solution: Fort McMurray to Yellowknife, to Edmonton and then to Fort Simpson
c. What is the busiest airport other that Edmonton?
Solution: Fort McMurray (four flights in, only one out each week)
Exercise:
The food chain is an integral part of our ecosystem.
1. Draw a digraph to represent the food web described below:
Black Bears eat plants, insects, carrion and fish
Fish eat insects and worms
Eagles eat fish, mice, squirrels and carrion
Insects, squirrels and mice eat plants
Spiders eat insects and other spiders
2. A company mail clerk delivers letters and packages to several departments throughout the day as indicated in the digraph below. Construct an adjacency matrix to represent the digraph.
Adding and Subtracting Matrices
Addition of matrices can only be done if the matrices are of the same dimensions. To add the two matrices, the corresponding elements are added.
Example 1:
Given: and
Solution 1:
Then: A + B =
A + B =
A + B =
There are two points to keep in mind about matrix addition:
(1) Matrix addition is commutative: A + B = B + A.
(2) Matrix addition is also associative: (A + B) + C = A + (B + C).
Example 2: Commutative
Given: and
Prove that A + B = B + A.
Solution 2:
Example 3: Associative
Given: , , and
Prove that A + (B + C) = (A + B) + C.
Solution 3:
Subtraction of matrices is defined in terms of addition: A – B = A + (-B). Two matrices can be subtracted only if they have the same number of rows and the same number of columns. To subtract one matrix from another, subtract their corresponding elements using the same procedure as with addition.
Example 4:
Given: and
Solution 4:
Then: S - T =
Example 5:
Given: and
Solution 5:
Then: A - B =
A + (-B) =
A + (-B) =
Exercises:
1. Add the following:
2. Subtract the following:
Multiplication of Matrices
Multiplication of a matrix by a real number is called scalar multiplication. A scalar can be defined as any single constant, variable, or expression. Scalar multiplication multiplies every element of the matrix by the real number.
Example 6: Given: , calculate 4J.
Solution 6:
Exercises: Express as a single matrix
1.
2.
3.
4.
5.
Multiplication of two matrices can only be performed if the first matrix has the same number of columns as the number of rows in the second matrix. The product matrix will have dimensions equal to the number of rows of the first matrix by the number of columns of the second matrix. The product of two matrices is done by row by column multiplication. The elements of the product matrix are obtained by adding the products of the elements of a row of the first matrix with the elements of a column of the second matrix.
In general:
and A = 2 × 2 B = 2 × 2
You can see that multiplication is row by column. If your students have difficulty performing this task, you could have them write, on a small piece of paper, the first row of the first matrix as a column. Now they can slide the elements of this column adjacent to the elements of each of the columns of the second matrix. This will provide your students with a visual of the numbers being multiplied. Students will repeat this for each row of the first matrix. (Note: Post-it-notes are great for this).
Example 7: Given:
Solution 7:
Example 8: Write each product as a single matrix:
a. b.
c. d.
e.
Before you apply matrices to real-life problems, you should have the students perform mathematical operations on matrices.
Exercises:
6. State the dimensions of each matrix.
- b. c. d.
e.
7. State the indicated sum, difference, or product.
- b.
c. d.
e. f.
Real-Life Problems Using Matrices
Example 1: The Fancy Fresh Fruit Company supplies cases of oranges, grapefruit and lemons to the local grocery store. For the month of January, Store A ordered 70 cases of oranges, 30 cases of grapefruit and 10 cases of lemons. Store B ordered 60 cases of oranges, 20 cases of grapefruit and 10 cases of lemons. Store C ordered 80 cases of oranges, 40 cases of grapefruit and 15 cases of lemons. Oranges cost $30/case: grapefruit cost $20/case and lemons cost $15/case.
Display the above order information as a table and then as a matrix.
a. In Matrix A, what does the element 20 tell you?
Solution: It tells you that Store B ordered 20 cases of grapefruit.
b. Is the element (1,3) mean the same as element (2,3)?
Solution: Element (1,3) represents 10 cases of lemons ordered by Store A.
Element (2,3) represents 10 cases of lemons ordered by Store B.
c. If you deleted row three from Matrix B, what information would you be eliminating?
Solution: You would be eliminating the cost of a case of lemons.
d. If you deleted column two from Matrix A, what information would you be eliminating?
Solution: You would be eliminating the number of cases of grapefruit ordered by each
store.
e. How many cases of fruit were ordered by the three stores?
f. How much money is owed to the Fancy Fresh Fruit Company by each store?
Store A owes $2850
Store B owes $2350
Store C owes $3425
Ornaments / #sold by Grade 10 / #sold by Grade 11 / #sold by Grade 12China dog / 20 / 35 / 28
China cat / 15 / 19 / 22
China bird / 10 / 16 / 12
China horse / 6 / 10 / 8
Crystal dog / 10 / 6 / 10
Crystal cat / 12 / 7 / 13
Crystal bird / 15 / 11 / 21
Crystal horse / 2 / 1 / 4
Example 2: The grade ten, grade 11 and grade 12 students at the local high school are all participating in a fundraiser. They are selling china and crystal ornaments. Four ornaments are available: dogs, cats, birds, and horses. The sales of the ornaments for each grade are listed here:
i) Represent the ornament sales for each grade in a matrix.
Solution:
Grade 10 Grade 11 Grade 12
ii) What does the first row in each matrix represent?
Solution: The first row represents the number of china ornaments sold.
iii) How many of each ornament were sold by the three grades?
Solution:
iv) How many more of each ornament did the grade 12 students sell than the grade 10 students?
Solution:
Exercise:
The following table represents the number of cases of Tylenol, Advil and Aspirin ordered from a drug company by four different stores.
Tylenol cost $120/case; Advil cost $100/case and Aspirin costs $90/case.
- Display the order as a matrix.
- Write the cost/case as a column matrix.
- Use matrix multiplication to determine the total amount owed to the drug company by each store.
Entering a Matrix on a Graphing Calculator
Example 1:
Input the two data matrices and display.
Step 1: On a graphing calculator press 2nd (x-1)Matrx, choose EDIT and
ENTER.
Step 2: By Matrix A, choose the dimensions of the matrix you are entering
(default is 1 x 1) . The matrix size will change on the screen to accommodate your
data (2 x 3). Remember to use the negation symbol (-) and not the
subtraction sign “ –“ .
Step 3: Press ENTER after the dimensions have been added.
Step 4: Input the data and press
ENTER after each entry.
Step 5: To enter the second matrix press 2ndMatrx, choose EDIT.
Step 6: Scroll down to Matrix B and ENTER.
Step 7: Input data for Matrix B (2 x 2)
You may add up to ten matrices as needed before you need to delete.
To check on the home screen press 2ndMatrx and scroll to the matrix you would like to see on the screen. Press ENTER and the desired matrix should be on the home screen. Press ENTER again and the elements will be listed.
To clear or delete a matrix, choose 2nd(+)MEM, select Mem Mgmt/Del (#2).
Scroll down to Matrix (#5), press ENTER. The dimensions and names of all entered matrices will be listed: there are two, (A) and (B). Scroll to any one (B) and press DEL (delete), not CLEAR, to remove from memory: (B) will be deleted in this case. You can also input data over an existing matrix to correct or replace.
Use a graphing calculator to solve the following:
Example 2:
Solution 2: