Chapter 7
Matrices
T. Iverson/Prange/Zurek
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Learning Targets and Homework Assignments
Learning Target
Practice for the
Learning Target
7.4.1
I can determine the dimensions of a matrix.
7.4 & 7.5 Wksht.
Pages 8-9
7.5.1
I can add and subtract 2 matrices. I understand
how dimensions of the matrices may make the
operation impossible. I understand how the
commutative property may apply to the
operation.
7.4 & 7.5 Wksht.
Pages 8-9
7.5.2
I can perform scalar multiplication on a matrix.
7.4 & 7.5 Wksht.
Pages 8-9
7.5.3
7.5.4
7.8.1
7.8.2
7.8.3
Review
I can multiply matrices. I understand how
dimensions of the matrices may make the
operation impossible. I understand how the
commutative property may apply to the
operation.
I can multiply 2 matrices in a real world
situation. I understand how dimensions of the
matrices may make the operation impossible. I
understand how the commutative property may
apply to the operation.
I can solve a system of equations using a
matrix.
I recognize when a system of equations has 0,
1, or infinitely many solutions using a matrix.
I can find the solution to an application
problem (2 or 3 variables) using a matrix.
Score on
Learning
Target
Quiz
Help
Needed?
Yes/No
7.5 Worksheet
Page 12-13
7.5 Worksheet
Page 12-13
7.8 Worksheet
Pages 17-18
7.8 Worksheet
Pages 17-18
7.8 Worksheet
Pages 17-18
7.8 Worksheet
Pages 20-23
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Essential Questions for the chapter
1. How do geometric relationships help us to solve problems and make sense of our world?
2. How do we use math models to describe relationships?
Essential Questions for the course
1. How is this similar or different from what I have done before?
2. What can I do to retain what I have learned?
3. Does my answer make sense? If not, what do I do?
4. Do I need help, and where do I go to find it?
5. How would a calculator make this problem easier to do?
6. How do I explain or justify my work to myself and others?
7. What is the given information and how do I use it?
4
LEARNING TARGET QUIZ SCORING RUBRIC
A+
MASTERY (+)
100% I completely understand the strategy and mathematical operations to be used, and I used them
or
correctly.
 I did all of my calculations correctly.
4.0




My work shows what I did and what I was thinking while I worked the problem.
The way I worked the problem makes sense and is easy for someone else to follow.
I followed through with my strategy from beginning to end.
My work was clear and organized.
M92%
or
3.7
MASTERY (-)
I completely understand the strategy and mathematical operations to be used, but one minor
error kept me completing the problem correctly.
DM
85%
or
3.4
DEVELOPING MASTERY
I understand the strategy and mathematical operations to be used, but a few minor errors kept
me from completing the problem correctly.



My thought process was correct but one minor error kept me from getting the correct answer,
BUT:
o The way I worked the problem makes sense and is easy for someone else to follow.
o I followed through with my strategy from beginning to end.
o My work was clear and organized.
I understood the concept, but my work lacks a few minor elements that would have made my
thought process easy for anyone to follow.
My thought process was correct but a few minor errors kept me from getting the correct
answer.
BU
75%
or
3
BASIC UNDERSTANDING
I used mathematical operations and a strategy that I think works for most of the problem.
IU
50%
or
2
INCOMPLETE UNDERSTANDING
I wasn’t sure which mathematical operations to use, and my plan didn’t work.
NE
0%
or
0
NO EVIDENCE
I did not demonstrate any understanding of the concept.











My work included an obvious conceptual mistake.
Several elements need to be added for my work to be easy to follow.
I know which operations I should have used, but couldn’t complete the problem.
I’m not sure how much detail I need in order to help someone understand what I did.
I made several significant calculation errors.
I tried several things related to the learning target(s), but didn’t get anywhere.
I was not able to reach an answer.
I left the problem blank.
I didn’t know how to begin.
I don’t know what to write.
I wrote down information not related to the learning target(s).
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Date _______
Notes: LTs 7.4 and 7.5
Warm up:
1a) 7 + 4 =
1b) 4 + 7 =
These are examples of the ____________________________________________________.
Essential Questions:
1. How do geometric relationships help us to solve problems and make sense of our world?
2. How do we use math models to describe relationships?
Learning Targets:
7.4.1
I can determine the dimensions of a matrix.
7.5.1
I can add and subtract 2 matrices. I understand how the dimensions of the matrices may make the operation
impossible. I understand how the commutative property may apply to the operation.
7.5.2
I can perform scalar multiplication on a matrix.
Vocabulary:
Matrix:
Element/Entry:
Dimensions:
Adding/Subtracting Matrices:
 2 3
A

 2 1
 1 5
B

 3 5 
By Hand
1 
C 
4
Calculator Steps
1. A  B 
Using your calculator find:
2. A  B 
3. A  C 
4. B  A 
5. B  A 
6. What observations can you make for questions 1 through 5?
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 2 5 0 
A

 3 5 1 
Scalar Multiplication & Naming:
1 5 9 
B

0 4 2
1. 2A 
2. Verify the answer above on your calculator. Is it the same?
3. Then try 3B on your calculator.
4. Also try 4 A  5B on your calculator.
5. What element is in A2,2?
6. What element is in B3,2?
7. A company offers three types of health care plans with two levels of coverage to its
employees. The current annual costs for these plans are represented in the table. If the annual
costs are expected to increase by 4% next year, what will be the annual increase for each plan
and what are the annual costs for each plan next year?
Plan
Coverage Level
Single
Family
Premium
694
1725
HMO
451
1187
HMO Plus
489
1248
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Homework: LTs 7.4.1, 7.5.1 & 7.5.2 Worksheet
:
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Homework: LTs 7.4.1, 7.5.1 & 7.5.2 Worksheet Continued
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Date _______
Notes: Lt 7.5
Warm ups:
1a) 3 x 4 =
1b) 4 x 3 =
These are examples of the ______________________________________________________.
Essential Questions:
1. How do geometric relationships help us to solve problems and make sense of our world?
2. How do we use math models to describe relationships?
Learning Targets:
7.5.3
I can multiply matrices. I understand how the dimensions of the matrices may make the operation impossible. I
understand how the commutative property may apply to the operation.
7.5.4
I can multiply matrices in a real world situation. I understand how the dimensions of the matrices may make the
operation impossible. I understand how the commutative property may apply to the operation.
Multiplying Matrices:
1 4 2 
A

0 1 2 
1 0 3 
B   0 1 2 
 4 2 1
1
C   2 
 3 
Dimensions:
1. Find AC by hand, then verify on your calculator:
2. Find CA with your calculator:
3. Observations:
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You Try:
1 2 0
D   1 4 5 
 0 3 2 
 4 1 1 
E   0 2 3
 4 2 3 
F  1 2 9
Evaluate with your calculator:
4. DF
5.
DE
6. ED
Location
Farm 1
Farm 2
Farm 3
Income
Peaches
165
243
74
Fruit Farms
Apricots
217
190
150
Plums
430
235
198
Apples
290
175
0
Peaches
Apricots
Plums
Apples
26
18
32
19
a) Write matrix A so that it represents the location/production table.
b) Write matrix B so that it represents the income by fruit table and so that it can be
multiplied by matrix A.
c) Calculate the total income for each farm.
d) Find the total income of all three farms.
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Homework: LTs 7.5.3 & 7.5.4 Practice Worksheet
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Homework: LTs 7.5.3 & 7.5.4 Worksheet Continued
10.
Menu Items
Location
Great America
Key Lime Cove
Wilderness Lodge
Income
Soft Pretzels
150
237
160
Cotton Candy
117
160
0
Popcorn
410
215
178
Hot Dogs
490
275
188
Soft Pretzels
Cotton Candy
Popcorn
Hot Dogs
6.50
5
5.50
6.95
a) Write matrix A so that it represents the location/production table.
b) Write matrix B so that it represents the “income by menu item” table and so that it can
be multiplied by matrix A.
c) Calculate the total income for these menu items at each amusement park.
d) Find the total income for these menu items for all 3 amusement parks.
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Date _______
Notes: 7.8
Warm up:
4 x  y  10
5 x  2 y  6
1. Solve this system of equations using elimination: 
2a. If you were to solve a system of equations by graphing, where is the solution located on the
graph?
2b. How does a graph look when there is no solution to the system of equations?
2c. How does the graph look when there is an infinite number of solutions to the system of
equations?
Essential Questions:
1. How do geometric relationships help us to solve problems and make sense of our world?
2. How do we use math models to describe relationships?
Learning Targets:
7.8.1
I can solve a system of equations using a matrix.
7.8.2
I recognize when a system of equations has 0, 1, or infinitely many solutions using a matrix.
7.8.3
I can find the solution to an application problem (2 or more variables) using a matrix.
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Solving Systems of Equations in Three Variables:
Example 1 Solve this system of equations using matrices:
3 x  y  z  6

2 x  y  2 z  8
4 x  y  3 z  21

Step 1: Enter the augmented matrix.
Step 2: Solve the system using “rref” (reduced row echelon form).
Step 3: Interpret your findings:
You Try: Solve the systems below:
1.
2 x  y  2 z  15

y  z  3  x
3x  y  18  2 z

2.
4 x  4 y  2 z  8

3x  5 y  3z  0
2 x  2 y  z  4

Ans: _______________
Ans: _______________
Interpretation:
Interpretation:
3.
2 x  3 y  8 z  10

z  4 y  1
2 x  3 y  8 z  5

Ans: _______________
Interpretation:
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7.8.3 I can find the solution to an application problem (2 or more variables) using a matrix.
The Laredo Sports Shop sold 10 balls, 3 bats, and 2 bases for $99 on Monday. On Tuesday they sold
4 balls, 8 bats, and 2 bases for $78. On Wednesday they sold 2 balls, 3 bats, and 1 base for $33.60.
What are the prices of 1 ball, 1 bat, and 1 base?
First define the variables.
Translate the information in the problem into three equations.
Set up your augmented matrix and interpret the results in context.
You try:
At the arcade, Ryan, Sara and Tim played video racing games, pinball, and air hockey. Ryan
spent $6 for 6 racing games, 2 pinball games, and 1 game of air hockey. Sara spent $12 for 3
racing games, 4 pinball games, and 5 games of air hockey. Tim spent $12.25 for 2 racing games,
7 pinball games, and 4 games of air hockey. How much did each of the games cost?
Use the process outlined above.
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Homework:7.8 Worksheet
Lt 7.8.1 I can solve a system of equations using a matrix.
Lt 7.8.2 I recognize when a system of equations has 0, 1, or infinitely many solutions using a matrix.
1.
2 x  3 y  z  0

 x  2 y  4 z  14
3x  y  8 z  17

3.
2 x  y  4 z  11

 x  2 y  6 z  11
3 x  2 y  10 z  11

2.
x  2 y  z  8

2 x  y  z  0
3x  6 y  3z  24

4.
2 x  4 y  z  10

4 x  8 y  2 z  16
3x  y  z  12

Learning Target 7.8.3 I can find the solution to an application problem (2 or more variables) using a
matrix.
5. Carly is training for a triathlon. In her training routine each week, she runs 7 times as far as
she swims, and she bikes 3 times as far as she runs. One week she trained a total of 232 miles.
How far did she run that week? Use the process outlined in your notes.
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6. AIRLINE TICKETS Last Monday at 7:30 A.M., an airline flew 89 passengers on a commuter
flight from Boston to New York. Some of the passengers paid $120 for their tickets and the rest
paid $230 for their tickets. The total cost of all of the tickets was $14,200. How many
passengers bought $120 tickets? How many bought $230 tickets?
7. SPORTS Alexandria High School scored 37 points in a football game. Six points are
awarded for each touchdown. After each touchdown, the team can earn one point for an extra
kick or two points for a 2-point conversion. The team scored one fewer 2-point conversions than
extra kicks. The team scored 10 times during the game. How many touchdowns were made
during the game?
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Warm Up before beginning the Chapter 7 Review Sheet
Two softball teams submit equipment lists to their sponsors.
Women's Team
Bats
Balls
Gloves
12
45
15
Men's
Team
15
38
17
Each bat costs $48, each ball costs $4, and each glove costs $42. Use matrices to find the total cost of
equipment for each team.
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Chapter 7 Test Review
You may use a graphing calculator on this test review.
7.4.1 I can determine the dimensions of a matrix.
Name the dimensions of the following matrices.
4
1.  8 
 1
1. _______________
 2 3 1 
2. 

3 0 4
2. _______________
7.5.1 I can add and subtract 2 matrices. I understand how dimensions of the matrices may make the operation
impossible. I understand how the commutative property may apply to the operation.
7.5.2 I can perform scalar multiplication on a matrix.
7.5.3 I can multiply 2 matrices. I understand how dimensions of the matrices may make the operation impossible. I
understand how the commutative property may apply to the operation.
Use the following for problems #3-5.
9 0 8
A  

5 2 4
3 5

D  
4 2 
4
B  
2
1 

3
4 1


E  1 5 


2 6 
0

C  5

2
1

3 

4
2 4 


F   7 3


 3 1 
Find each of the following. If the operation is not possible, write  .
3.
2C + 3F
3. ________________
4.
4BF
4. ________________
5.
AE
5. ________________
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7.5.4
I can multiply matrices in a real world situation. I understand how dimensions of the matrices may make
the operation impossible. I understand how the commutative property may apply to the operation.
Use the tables for the questions 6-9 below:
July
August
September
October
Poster
$5.99
White Stripes
Poster
45
63
35
24
SALES
A.A. Bondy
T-shirt
34
24
34
75
PRICE PER ITEM
T-Shirt
$7.99
Ryan Adams
CD
54
64
65
25
CD
$9.99
6.
Write a matrix for the SALES table.
7.
Write a matrix for the PRICE PER ITEM table.
8.
Multiply the 2 matrices and write the solution matrix for the situation.
9.
What does each solution represent?
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7.8.1 I can solve a system of equations using a matrix.
7.8.2 I recognize when a system of equations has 0, 1, or infinitely many solutions using a matrix.
10.
Solve using a matrix.
10.________________
2 x  y  5

 x  3 y  13
11.
In the space below, describe what the solution for #8 represents graphically?
12.
Solve this system of equations.
12. ________________
2 x  y  4 z  6

 x  5 y  2 z  6
3x  2 y  6 z  8

13. Solve this system of equations.
13. ________________
 x  2 y  3z  6

2 x  4 y  6 z  12
3x  6 y  9 z  18

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7.8.3 I can find the solution to an application problem (2 or 3 variables) using a matrix.
14.
Alec is training for his pilot’s license. Flight instruction costs $105 per hour, and the
simulator costs $45 per hour. The school requires students to spend 4 more hours in airplane
training than in the simulator. If Alec can afford to spend $3870 on training, how many hours
can he spend training in an airplane and in a simulator?
Write a system of two equations that describe this situation and solve.
Define your variables. ___________________________
Equation 1 _____________________________________
Equation 2 _____________________________________
Solution ____________________________________________________
15.
The Harvest Nut Company sells made-to-order trail mixes. Stephanie’s favorite mix
contains peanuts, raisins, and carob-coated pretzels. Peanuts sell for $3.20 per pound, raisins are
$2.40 per pound, and the carob-coated pretzels are $4.00 per pound. Stephanie chooses to have
twice as many pounds of pretzels as raisins, wants 5 pounds of mix, and can afford $16.80. How
many pounds of peanuts, raisins, and carob-coated pretzels can Stephanie buy?
Write a system of three equations that describes this situation and solve.
Define your variables. __________________________
Equation 1 _____________________________________
Equation 2 _____________________________________
Equation 3 _____________________________________
Solution ____________________________________________________
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