Discrete Math Through Applications (DMTA) Lesson 1
A MATRIX is a rectangular arrangement of numbers by rows and columns used to display, organize and manipulate information. Brackets [ ] are used to enclose the numbers of the matrix.
Example Matrix: Suppose there are four pizza restaurants in your town. The table below shows the prices each restaurant charges for a large one-topping pizza, a one-liter bottle of soda, and a family-size order of salad.
From the table we are able to create the matrix A to describe the pizza restaurant prices.
Gina’s Vinny’s Tony’s
Pizza $12.16
Drink $1.15
Salad $4.05
$10.10
$1.09
$3.69
$10.86
$0.89
$3.89
Sal’s
$10.65
$1.05
$3.85

A





12
1 .
.
16
15
4 .
05
10
1
3
.
.
.
10
09
69
10
0
3
.
.
.
86
89
89
10
1
3 .
.
.
65
05
85




Each number in the matrix is called an ELEMENT (or entry) of the matrix. Individual elements in a matrix are identified by row number and column number location in that order. For example, the value 10.65 is the element in row 1 and column 4 of matrix A and is identified as A
14 or A
1,4
.
1. What is the value of element: A
21
? ____ ______ A
12
? __________ A
34
? ___ _____ ____
2. Identify the element for the value: 12.16. _______ 1.05 ______ 3.69 ________
The DIMENSION (or order) of a matrix is determined by the number of rows and columns in the matrix listed in that order. In general, a matrix with m rows and n columns is called an m
 n matrix.
3. What is the dimension of this restaurant matrix?
4. How many elements are in the matrix A?
5. How many elements does each of the following matrices have?
2

8 matrix: _____ 8

2 matrix : ______ 7

1 matrix : ______ 3

3 matrix: ______
COLUMN MATRIX is a matrix with only one column. Example: If you create a matrix of only
Sal’s menu items, the result is a 3

1 column matrix.
Write the column matrix for Vinny’s.




10
1 .
.
65
05
3 .
85




ROW MATRIX is a matrix with only one row.
Example: If you create a matrix of only the pizza prices of the four restaurants, the result is a 1

4 row matrix.

12 .
16 10 .
10 10 .
86 10 .
65

Write the row matrix for salad prices.
6. A garment company receives orders from three clothing shops. Shop 1 orders 25 jackets, 75 shirts, and 75 pairs of pants. Shop 2 orders 30 jackets, 55 pairs of pants, and 45 shirts. Shop 3 orders 40 shirts, 20 jackets, and 35 pairs of pants. Display this information in a matrix with columns as jackets, shirts, and pants.
7. For breakfast, Yoko had cereal, a banana, a cup of milk and a slice of toast. She recorded the nutritional information in her food journal: cereal has 165 calories, 33 g of carbohydrate, 3 g of fat, and no cholesterol; a banana has 120 calories, 26 g of carbohydrate, no fat, and no cholesterol; milk has 120 calories, 11 g of carbohydrate, 5 g of fat, and 15 g cholesterol; and toast has 125 calories, 14 g of carbohydrate, 6 g of fat, and 18 g cholesterol. Display this nutritional information in a matrix N whose rows represent the foods and columns represent calories, carbs, fat, and cholesterol in that order.

Discrete Math Through Applications (DMTA) Lesson 2
Additional Key Terms:
Entries in a square matrix located in row i , column j where i = j are said to be located on the MAIN
DIAGONAL (top left corner to bottom right corner).
A square matrix S with dimension n
 n is
SYMMETRIC if S ij
= S ji
where i and j = 1, 2, 3,…, n .
(Matrix is a reflection across the diagonal)
1.
Is matrix K or matrix L symmetric?
2.
The elements and values of the main diagonal for matrix K are K
11
= 0, K
22
= –1, and K
33
= 4. Identify the elements and the values of the main diagonal in matrix L.
K





0
2
7
2

1

5
7


5
4


L




3
7




8
1
7

5
4
4

1
4
6

2

8
4
0
2






MATRIX ADDITION AND SUBTRACTION: In general, you can add or subtract matrices only if they have the SAME DIMENSIONS.
The resulting matrix will have the same dimensions as the original matrices. a.



2
3
4
5






7
1

2
8


Examples: Identify which operations have a solution or not. b.


1
0
5
2
7
4





1


5
7
0


2
4


c.


1
0
5
2
7
4





3
5 6
0

 d.




1
3
5
The operations of addition and subtraction are by corresponding elements of the matrices:
0
0
0










1
2
3

5
0
4




If A and B are matrices of the same dimensions, then the sum A + B is formed by adding the corresponding elements of A and B . The difference A – B is formed by subtracting the corresponding elements of B from the corresponding elements of A .


10
11
15
8
13
14





7
9
10
7
12
10


Addition Example:



10
11


7
9
15
8


10
7
13 
14 
12
10





17
20
25
15
25
24




-
2
3
3.5
4





-
7
1
Subtraction Example:
0.5
2





2
-
(-1)
3 7
3.5
4 -
-
2
.5


SCALAR MULTIPLICATION: In general, scalar



-
3
10
3
2


Scalar Multiplication Example: multiplication is a process in which we multiply each element in a matrix A by a constant number k and k is referred to as the scalar. The resulting matrix is kA .
3


-


3
6
2
-
1
5
3









3
3
3



3
-2
6 3
3
3


 1
5
-3









-
9
6
18
3
15


9


MATRIX CALCULATOR COMMANDS:
STEP 1: Making a matrix

Create the matrix: [2 ND ]

[Matrix]

EDIT

Scroll to a matrix

[ENTER]

Enter the correct dimensions of the matrix: Rows X Columns

Fill in elements in the correct locations.

Quit to Main Screen: “[2 nd ]

[MODE]”
STEP 2: Using the matrix to do operations

Put Scalar multiplication before a matrix. Otherwise add or subtract like number

Pick a matrix to use in operation : [2 ND ]

[Matrix]

NAMES

Scroll to the matrix

[ENTER]

1)
A



8
2
MATRIX OPERATION PRACTICE: Try by hand, and then check with calculator
6
9
7
3

 , B



3
5

7
4

2
1

 ,and C





1
2
3
.
5
8

5
.
5
4



 a) Find A – B b) Find 4C d) Find 2A + B c) Find C + A
2)
L




2
3
5
4

 , M



2
7


3
2

 , and N




1
0
0

5 
 b) Find L + 2N c) Find N – 2M a) Find L + N – M d) Find 3L + 2M
3) Find the values of x, y, and z for

 x
2 e) Find 2(L + M)

3
1





2 .
5
5

11 y





10
7
 z
3

 f) 2M – 3N + 5L
4) Find the values of a, b, and c for


 a
2
11
8





1
2
 b
4






6
4
 c
5 .
5


5) The tables below show the statistics for the 2003 National League batting leaders and their statistics in the following year of 2004. Create a matrix that shows the change in their statistics from 2003 to 2004.
2003 AB R H HR RBI Avg.
Pujols 591 137 212 43 124 .359
Helton 583 135 209 33 117 .358
Bonds 390 111 133 45 90 .341
2004 AB R H HR RBI Avg.
Pujols 592 133 196 46 123 .331
Helton 547 115 190 32 96 .347
Bonds 373 129 135 45 101 .362
6) The tables show the distance Jane and Jessica traveled (in miles) during a two-week period on the
Appalachian Trail. Write a matrix that shows the total miles driven for both weeks for both women.
Jane M T W T F Jessica M T W T F
Week 1 45 60 50 23 15
Week 2 25 30 35 42 47
Week 1 32 35 33 35 36
Week 2 47 42 40 39 31

Discrete Math Through Applications (DMTA) Lesson 3
Fact 1: Matrix multiplication is a process of pairing elements from the row of the first matrix and the column of the second matrix to multiply all pairs and then add those products.
Fact 2: The number of columns in the first matrix must be equal to the number of rows in the second matrix of the multiplication.
Fact 3: The resulting dimension of the product matrix will have the number of the rows of the first matrix and then number of columns of the second matrix.
x
In general, if A is a 1
 k row matrix and B is a k

1 column matrix, then A

B is a 1

1 single value matrix.
Example: Suppose Ruben stops at the convenient store on his way to school to by snacks. He buys four bags of chips at 30 cents each, five candy bars at 35 cents each, a box of cheese crackers for 50 cents, three packs of sour drops at 20 cents each, and two bags of cookies at 75 cents each.
Ruben’s purchases can be displayed in a
1

5 row matrix Q (quantity matrix) .
The prices of each snack can be displayed in a 5

1 column matrix P (price matrix) .
Q


4 5 1 3
P


0 .

 0 .




0
0
0
.
.
.
30

35


50


20
75


2

If we want to find the total amount Ruben paid for his snacks, we can multiply the price matrix P by the
quantity matrix Q.
QP


4 5 1 3 2








0 .
30

0 .
35


0
0
0 .
.
.
50
20
75





Practice #1:
Suppose Ruben’s friend, Terri, goes along with him to the store. Terri buys a bag of chips for 25 cents, two candy bars at 45 cents each, two packs of gum at 30 cents each, and a medium drink for
75 cents. a. Create Terri’s quantity matrix Q and Terri’s price matrix P for purchases. b. Use matrix multiplication to find the total cost QP of Terri’s purchases.
Practice #2: Mr. Dunn has $10,000 in a 12-month CD at 7.3% (annual yield), $17,000 in a credit union at
6.5%, and $12,000 in bonds at 7.5%. What will the value of his investments be after one year? a. Write a row matrix V that shows the current value of each investment. b. Write a column matrix Y that shows the yield of each investment. c. Use matrix multiplication to find the value of Mr. Dunn’s earnings after one year

If A is a
1
 k
x
row matrix and B is a

multi-dimensional matrix, then A

B is a
1
 n
row matrix.
Example: The matrix T that shows the cost of ordering pizzas with two toppings and salads with a choice of two dressings at 3 pizza restaurants. Suppose for a pizza party, you decide to order 5 two-topping pizzas and 3 salads with two dressings and matrix A shows the number of pizzas and salads you would like to order.
T

Pizza
Salad
Vinny ' s


11
3 .
.
25
69
Tony ' s
11
4 .
.
96
34
Sal '
11
4 .
.
90
35

 s
A


5 3

To find each restaurant’s total cost of order by multiplying the 1

2
AT


5 3



11 .
25
3 .
69
11
4 .
.
96
34
11 .
90


4 .
35
row matrix A and 2

3 matrix T and
AT


5 3
 

11
3 .
.
25
69
11 .
96
4 .
34
11
4 .
.
90
35


1. Write an interpretation of AT
12
.
2. Given the dimensions for matrices Q and P, state (Yes or No) whether the product QP is defined (does it exist?). If it is defined, give the dimensions of the product. a. Q: 1

4 , P: 4

2 defined? ________ dimension of QP __________ b. Q: 1

3 , P: 2

3 defined? ________ c. Q: 1

5 , P: 5

4 defined? _________ dimension of QP __________ dimension of QP __________ d. Q: 1

2 , P: 4

2 defined? ________ dimension of QP __________
3. The table shows a credit union’s investments in loans and bonds in
Massachusetts, Nebraska and California (amounts are in thousands of dollars). The current yields on these investments are 6.5% for loans and 7.2% for bonds. Use matrix multiplication to find the total earnings for each state. Show the complete matrix equation.
4. Use A



3
2
8
0

4
1

 B


2 4

C

Loans
Bonds
MA
230
780


2
4

 to compute the given expressions.
NE
440
860
CA
680
940 a. 3A b. BA c. BC

x
Matrix Key Term: An IDENTITY MA
TRIX (I) , is any square matrix in which each entry along the main diagonal is 1 and all other entries are 0. If matrix I is an identity matrix, then for any A matrix IA = AI = A.
In order for the product of two matrices to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. In general, if A is a m
 k row matrix and B is a k
 n multi-dimensional matrix, then A

B is a m
 n row matrix.
Example: Recall the matrix T and suppose for your pizza party, you are considering three different options for pizza and salad combination shown in the 3

2 matrix B .
T

Pizza
Salad
Vinny ' s


11
3 .
.
25
69
Tony ' s
11
4 .
.
96
34
Sal ' s
11 .
90
4 .
35


B

Option # 1
Option # 2
Option # 3
Pizza




4
4
5
Salad
3
4
3




The different combinations you are considering are. If we multiply matrix B times matrix T, the product will be a 3

3 matrix that will represent each restaurant’s cost for each option.
BT





4
4
5
3
4
3



 

11
3 .
.
25
69
11
4 .
.
96
34
11
4 .
.
90
35


BT






4
4
5
3
4
3 







11
3 .
.
25
69
11
4 .
.
96
34
11
4 .
.
90
35 


1. In the product matrix, BT
11
represents the cost of four pizzas and three salads at Vinny’s. Interpret
BT
23
and BT
32
.
2. Use
A





2
1
1
3 


,
B





2
4
1
3
0
5



,
C






1
0
1

2
4
2



 to compute the given expressions.
2a. AB
2c. BC
2b. BA
2d. CB

3. Matrix Multiplication Practice: If possible find the solution. a.


2
0
0
3





5
2
3
1

 b.

4

3 2






0
4
1

2
1
3

5
1
2



 c.


1
8
3
6
5
4
7
2







3
0
5
0
0
1
4

3

2
8
0
3




 d. 




2
1
3
5


0
4




7
2
4
9


8
3

 e. 




6
1
3

2
1
3

1
2
6








0
1
1

1
0
1


0
1
1



 f.




 
2 .
5
3 .
75

3
1
.
.
5
25
4

1
2
3
8
9
12
11


4
7 .
6
.
.
3
5
.
25

75
5








 
2
3
0
4





4. A hobby shop has three different locations in the North, South, and East. The store’s sales for July are shown in the table. Suppose that the model trains sell for $40 each, cars for $35, planes for $80 and trucks for $45.
4a. Set up and solve a matrix multiplication to
North South East find the total sales at each location.
Model Trains 10 8 12
4b. If the stores decided to sell trains for $45, cars for $40, planes for $65, and trucks for $25, would
Model Cars 6
Model Planes 3
Model Trucks 4
5
2
4
2 they have made more money?
3 2
5. The table gives the nutritional information for the menu items at a restaurant.
5a. Rosa orders a special, fries, and a shake.
Max has a cheeseburger, a baked potato, and
Calories Fat
(g)
Cholesterol
(mg) a shake. Set up an appropriate matrix Cheeseburger 450 40 50 multiplication for their orders and the Special 570 48 90 nutritional information to find the total Baked Potato 500 45 25 amount of national information for their French Fries 300 30 0 orders. Shake 400 22
5b. Daryl orders 2 cheeseburgers, a shake, and fries. Jordan orders a special, cheeseburger, baked potato, and two shakes. Set up an appropriate matrix
50 multiplication for their orders and the nutritional information to find the total amount of national information for their orders.

Population growth of animal populations is not constant as seen in Logistic Growth of Section 10.4, which modeled growth based on the available space in a habitat. Additionally, for animal populations it is important to understand how different age groups within a population contribute to the growth of the overall population.
If you know the age distribution of a population at a certain date and the birth and survival rates for age-specific groups, you can use this data to create a mathematical model, called the LESLIE MATRIX MODEL.
P. H. Leslie examined a population of an imaginary species of small brown rats. In order to simplify the model, the following assumptions are made.

Only the female population is considered

Birth rates and survival rates are held constant over time.

The survival rate of a rat is the probability that it will
Age
(months)
0-3
3-6
Birth
Rate
0
0.3
Survival
Rate
0.6
0.9 survive and move into the next age group.

The lifespan of these rodents is 15-18 months.

The rats will have their first litter at approximately 3 months and continue to reproduce every 3 months until they reach the age of 15 months.
6-9
9-12
12-15
15-18
0.8
0.7
0.4
0
0.9
0.8
0.6
0
Suppose the original female rat population is 42 animals with the age distribution below.
Age (months) 0-3 3-6 6-9 9-12 12-15 15-18
Number 15 9 13 5 0 0
#1: How do you find a new population distribution after 3 months (1 transition)?
1) How many new rats have been born?
2) How many rats were able to survive from their current age group to the next age group?
Age
(months)
0-3
#
15
Survival
Rate
0.6
Rats to move up age group
3-6 9 0.9
6-9
9-12
12-15
13
5
0
0.9
0.8
0.6
15-18 0 0
3) What is the total number of rats and write a distribution table?

Find the new population distribution after 6 months (2 transition)?
1) How many new rats have been born from the 1 st to 2 nd transition?
2) How many rats were able to survive from their current age group to the next age group?
# Rats to move up age group Age
(months)
0-3
Survival
Rate
0.6
3-6 0.9
6-9
9-12
12-15
15-18
0.9
0.8
0.6
0
3) What is the total number of rats and write a distribution table?
#2: A deer species has the following birth and survival rates for transitions of 2 years.
Age 0-2 2-4 4-6 6-8 8-10 10-12
Deer 50 30 24 24 12 8
Use the rates and the initial population distribution of
148 deer to calculate the number of deer after
2a. Find the number of newborn female deer after two years.
Age
(years)
0 – 2
2 – 4
4 – 6
6 – 8
8 – 10
Birth
Rate
0
0.8
1.7
1.7
0.8
10 – 12 0.4
2b. Calculate the number of deer that survive in each age group after two years.
2c. Find the population distribution after 4 years (2 cycles)
Survival
Rate
0.6
0.8
0.9
0.9
0.7
0

Discrete Math Through Applications (DMTA) Lesson 5
The Leslie Model is computationally tedious because you need to find separate values for each age groups survival and the overall number of newborns in any population to complete one cycle or transition of the population distribution. MATRIX MULTIPLICATION will simplify these tedious calculations by appropriately creating a matrix for the population distribution and a LESLIE MATRIX to find future population distributions.
The LESLIE MATRIX, L, is a SQUARE matrix that uses the birth and survival rates of a population such that the rows will always represent the different age groups of population. The first column will represent the birth rates of the age groups, and the following columns will contain the survival rate only for a specific age group and all other elements of that column will be zero to create the “super diagonal” of the matrix. Do not include the survival rate of the last age group because it is always ZERO as the max life span of the population.
L

AgeGroup
AgeGroup
...
1
2
LastAgeGro up






Birth
Birth
Birth
Group
Group
...
1
2
LastGroup
Survival
0
0
0
Group
Example #1: Leslie Model Rat Example from Lesson 4
1 0
Survival Group 2
0
0
0
0
...
0
Survival
0
0
2
0 ndtoLast






Age (months) 0-3 3-6 6-9 9-12 12-15 15-18
Number 15 9 13 5 0 0
Age
(months)
0-3
Birth
Rate
0
Survival
Rate
0.6 1a. Create a matrix multiplication P
0
L with a row matrix, P
0,
for the initial population distribution and the Leslie Matrix, L to find
3-6 0.3 the population after one cycle. Compare to Lesson 4 answer.
P
0
L


15 9 13 5 0 0








0
0

 0
0
0
.
.
.
3
8
.
7
0
4
0 .
6
0
0
0
0
0
0
0 .
9
0
0
0
0
0
0
0 .
9
0
0
0
0
0
0
0 .
8
0
0
0
0
0
0
0 .
6
0









6-9
9-12
12-15
15-18
0.8
0.7
0.4
0
1b. How might you calculate the next population distribution (2 cycles) using your Leslie Matrix?
0.9
0.9
0.8
0.6
0
In general if you know the initial population distribution, P
0
, and the Leslie Matrix, L, for a population, then how can you calculate any future population distribution into the future (N transitions or cycles)

1c. Find the population distribution 24 months or 8 CYCLES into the future from the initial population?
1d. Calculate P
25
, P
26
, and P
27 and their total populations.
1e. Calculate the rate of growth (percent of change) of TOTAL populations between these cycles.
Example #2: Leslie Model Deer Example from Lesson 4
Age 0-2 2-4 4-6 6-8 8-10 10-12
Age
(years)
Birth
Rate
Survival
Rate
Deer 50 30 24 24 12 8 0 – 2 0 0.6
2a. Set up the Leslie Matrix for the deer.
2b. Find the population distribution after 2 years (1 transition/ cycle). Check your answer with Lesson 4.
2c. Use matrix multiplication and your calculator to find the population distribution after 10 years (5 transitions or cycles)
2 – 4
8 – 10
0.8
4 – 6 1.7
6 – 8 1.7
0.8
0.8
0.9
0.9
0.7
10 – 12 0.4 0
Example #3: Suppose there is a certain kind of bug that lives at most 3 weeks and reproduces only in the third week of life. Fifty percent of the bugs born in one week survive into the second week, and 70% of the bugs that survive into their second week also survive into their third week. On average, six new bugs are produced for each bug that survives into its third week. A group of five 3-week-old female bugs decide to their home in the basement.
3a. Construct the Leslie matrix for this bug population.
3b. What is P
0
?
3c. Find P
3
.
3d. What is the total number of female bugs in P
5
?
3e. Approximately how long will it take for at least 1,000 female bugs to populate the basement.
Example #4: The characteristics of the female population of a herd of mammals are shown below.
Age Groups 0 – 4 months
4 – 8 months
8 – 12 months
12 – 16 months
16– 20 months
20 – 24 months
Birth Rate
0 0.5 1.1 0.9 0.4 0
Survival Rate
0.6 0.8 0.9 0.8 0.6 0
Initial Female
Population
22 22 18 20 7 2
4a. What is the expected life span of this mammal?
4b. Construct the Leslie matrix for this population?
4c. Calculate P
25
, P
26
, and P
27
.
Find the total population of each population distribution.
4d. Calculate the rate of growth (percent of change) of TOTAL populations between these cycles.
4e. What might be the long-term rate of growth for this population?

Discrete Math Through Applications (DMTA) Lesson 6
The Leontief Input-Output Model was developed by Harvard economist, Wassily Leontief, in the 1960s. He began his study by constructing input-output tables that described the flow of goods and services among various sectors of the economy in the United States.
Example #1 : Power Source is a company that manufactures batteries used to power electric motors.
However, not all the batteries produced by the company are available for sale outside the company. For every 100 batteries produced, three (3%) are used within the company.
In general, for a internally and P
TOTAL PRODUCTION

0 .
03 P
of P batteries by this company, 0 .
will be available for external sales to customers. If
03
D
P batteries will be used
represents DEMAND , the number of batteries available for external sales, then the number of batteries available for external sales is D

P

0 .
03 P
Suppose the company receives an order for 5,000 batteries. What must the total D

P

0 .
03 P production be to satisfy this external demand for batteries?
To find the total production necessary, substitute 5,000 for D in the previous equation and solve for P .
The company must produce 5,155 batteries to satisfy an external demand for 5,000 batteries.
5 , 000

5 , 000
P


P
P


1
0 .
03 P

0 .
03

5 , 155 batteries
Example #2: The High-Tech Computer Company produces computer chips. For every dollar’s worth of chips it produces for external sales, it uses 2 cents worth in its manufacturing process.
1a. Write an equation for High-Tech that represents external demand D in terms of total production P .
1b. What must the total production of chips be to meet an external demand of $20,000 worth of chips?
Example #3 - TWO-SECTOR ECONOMY: Suppose Power Source buys an electric motor company and begins producing motors as well as batteries. The company’s primary reason for this merger is that electric motors are needed to produce batteries. The production requirements for each division are described on the following page.
Battery Division
1. For the battery division to produce 100 batteries, it uses 3% of its own batteries.
2. For every 100 batteries produced, 1 motor is required from the motor division. (1% of total produced)
Motor Division
1. For the motor division to produce 100 motors, it must use 4% of its own motors.
2. For every 100 motors produced, 8 batteries are required from the battery division. (8% of total produced)
The production needs within this two-sector economy can be illustrated in the weighted digraph:.
The battery division requires 0.03
b batteries from its own division and 0.01
b motors from the motor division.
The motor division requires 0.04
m motors from its own division and 0.08
m batteries from the battery division.

Another way to view the company’s production needs is in a consumption matrix for the economy.
FROM
Battery
Motor
Battery


0 .
0
03
.
01
TO
Motor
0 .
08
0 .
04


Battery example : to produce 200 batteries, the battery division will need (0.03)(200) = 6 batteries from the battery division and (0.01)(200) = 2 motors from the motor division.
Motor example : To produce 50 motors, the motor division will need (0.08)(50) = 4 batteries from the battery division and (0.04)(50) = 2 motors from the motor division.
Example #4 - TWO-SECTOR ECONOMY: Suppose the High-Tech Computer Company adds another division that produces computers. Each division within the expanded company uses some of the other division’s products. For every dollar’s worth of chips it produces, the Chip Division requires 2 cents’ worth of computer chips and 1 cents’ worth of computers. For every dollar’s worth of computers it produces, the
Computer Division requires 20 cents’ worth of computer chips and 3 cents’ worth of computers. a. Construct and label a digraph and a consumption matrix for this economy. b. Suppose the Chip Division produces $1000 worth of chips. How much input does it need from itself and from the Computer Division. c. Suppose the Computer Division produces $5000 worth of computers. How much input does it need from itself and from the Chip Division.
PRACTICE PROBLEMS: Lesson 6
1. A utility company produces electric energy. Suppose that 5% of the total production of electricity is used up within the company to operate equipment needed to produce the electricity. Complete the following production table for this one-sector economy.
Total Units Used
Production Units
500 0 .
05
Internally
 

900
P
100
250
Unit for
500

0 .
05
External Sales
 

2,375
7,125

For questions 2-4, use the following consumption matrix for a company with two departments, Service and Production .
Service
Production
Service


0
0
.
05
.
04
Production
0 .
20
0 .
01


2. Draw a weighted digraph to show the flow of goods and services within this company.
3. Complete the following. a. For every dollar’s worth of output, the service department requires __________ cents’ worth of input from its own department and __________cents’ worth of input from the production department. b. For every dollar’s worth of output, the production department requires _________cents’ worth of input from its own department and ________cents’ worth of input from the service department.
4. Suppose that the total output for the service department is $20 million over a certain period of time and the total output from the production department is $40 million. a. How much of the total output for the service department is used within the service department?
How much input is required from the production department? b. How much of this total output for the production department is used within the production department? How much input is required from the service department? c. Combine the information in parts a and b to find how much of the total output from the service and production departments will be available for sales demands outside the company.
For questions 5 and 6, use the digraph below that represents the flow of goods and services in a two sector economy between transportation and agriculture.
0.15
A
0.35
T
0.10
0.30
5. Construct a consumption matrix for this two-sector economy.
6. If the total output for the agriculture sector is $50 million and the total output for the transportation sector is $100 million over a certain period of time, find the following. a. The total amount of agricultural goods used internally by this two-sector economy. b. The total amount of agricultural goods available for external sales. c. The total amount of transportation services used internally by this two-sector economy. d. The total amount of transportation services available for external sales.

Discrete Math Through Applications (DMTA) Lesson 7
Recall the consumption matrix of Power Source from lesson 6 :
C

Battery
Motor
Battery


0
0
.
03
.
01
Motor
0 .
08
0 .
04


Suppose we want to write equations that will help us to find the total production of batteries b and motors m necessary to meet internal requirements as well as external sales demands of 400 batteries and 100 motors.
The total batteries b produced, b

0 .
03 b

0 .
08 m

400
, equals the number of batteries used within the battery division (0.03
b ) plus the number of batteries sent to the motor division
(0.08
m ) plus the number of batteries required for sales outside the company (400).
The total number of motors m produced, m

0 .
01 b

0 .
04 m

100
, equals the number of motors sent to the battery division (0.01
b ) plus the number of motors used within the motor division (0.04
m ) plus the number of motors required for sales outside the company (100).
The general matrix equation can also be Gives you a system of equations

 b m


0 .
03 b
0 .
01 b


0 .
08 m
0 .
04 m


400
100 written :
P

CP

D
Total Production Matrix:
P


 b m


To solve for total production P,
P

CP

D
Consumption Matrix:
C



0
0 .
.
03
01
0 .
0 .
08
04


P

CP

D
Demand Matrix: D



400
100


IP
( I


CP
C ) P


D
D
This system written as a matrix equation:

 b m





0 .
03
0 .
01
0 .
08
0 .
04



 b m





400
100


( I

C )

1
( I

C ) P
P
2

2

( I

C )

1
identity matrix.

D
To find the total production of Power Source, solve

 b m







1
0
0
1





0
0 .
.
03
01
P

( I
0 .
08
0 .
04





1 

400
100




C )

1
D


421
108 .
.
3
6


( I

C )

1
D
, where I is the

PRACTICE PROBLEMS: LESSON 7
1. Recall the consumption matrix for High-Tech Computers:





1a. Suppose that the total production for the company is $40,000 worth of computer chips and $50,000 worth of computers. Write a production matrix P.
1b. Compute the matrix product CP to find the amount of each product that the company uses internally.
1c. Use the information from parts a and b and the matrix equation D

P

CP to compute the amount of computer chips and computers available for sales outside the company (external demand).
2. Recall the consumption matrix from Lesson 6.
C
 Service
Production
Service


0
0 .
.
05
04
Production
0
0
.
20
.
01


2a. Suppose that the total output for the service department of the company is $20 million and the total output for the production department is $40 million. Write a production matrix P.
2b. Compute the matrix product CP to find the output from each department that the company uses internally.
2c. Compute the output that is available for sales demands outside the company
(external demand).
2d. The company must meet external demands of $25 million in service and $50 million in products over a period of time. What must be the total production in service and products to meet this demand?
3. Consider the consumption matrix for the two-sector economy involving agriculture and transportation.
C
 Agricultur e
Transporta tion
Agricultur e


0
0
.
35
.
30
Transporta tion
0
0
.
15
.
10


Suppose the economy must meet external demands of $10 million in agricultural products and $15 million in transportation services. Find the total production of agriculture products and transportation services necessary to satisfy these demands.

USE A SEPARATE PIECE OF PAPER AS NEEDED.
1. A group of students is planning a retreat. They have contacted three lodges in the vicinity to inquire about rates. They found that Crystal Lodge charges $13.00 per person per day for lodging, $20.00 per day for food, and $5.00 per person for use of the recreational facilities. Springs Lodge charges $12.50 for lodging, $19.50 for meals, and $7.50 for use of the recreational facilities. Bear Lodge charges $20.00 per night for lodging, $18.00 a day for meals, and there is no extra charge for using the recreational facilities.
Beaver Lodge charges a flat rate of $40.00 a day for lodging (meals included) and no additional fee for use of the recreational facilities.
Display this information in a matrix C where the rows are the lodges. Label the rows and columns.
2. Your math club is planning a Saturday practice session for an upcoming math contest. For lunch the students ordered 35 Mexican lunches, 6 bags of corn chips, 6 containers of salsa, and 12 six-packs of cold drinks. Suppose that the club pays $4.50 per lunch, $1.97 per bag of corn chips, $2.10 for each container of salsa, and $2.89 for each six-pack of cold drinks.
2a. Set up a matrix multiplication to find the total cost. Identify the rows and columns of your matrices.
2b. Find the total cost.
3. Mr. Jones has been shopping for a vacuum-powered cleaning system. He found one at Z-Mart and another model at Base Hardware. The Z-Mart system cost $39.50, disposal cartridges were 6 for $24.50, and storage cases were $8.50 each. At Base Hardware the system cost $49.90, cartridges were 6 for
$29.95, and cases were $12.50 each.
3a. Write and label a matrix showing the prices for the three items at the two stores.
3b. Mr. Jones decided to wait and see if the prices for the systems would be reduced during the upcoming sales. When he went back during the sales, the Z-Mart prices were reduced by 10% and the Base Hardware prices were reduced by 20%. Construct a matrix showing the sale prices for each of the three items at the two stores.
3c. Use matrix subtraction to compute how much Mr. Jones could save for each item at the two stores.
4. An artist creates plates and bowls from small pieces of colored woods. Each plate requires 100 pieces of ebony, 800 pieces of walnut, 600 pieces of rosewood, and 400 pieces of maple. It takes 200 ebony pieces,
1200 walnut pieces, 1000 rosewood pieces, and 800 pieces of maple to make a large bowl. A small bowl takes 50 pieces of ebony, 500 walnut pieces, 450 rosewood pieces, and 400 pieces of maple.
4a. She currently has orders for five plates, three large bowls, and seven small bowls. Set up and solve a matrix multiplication to compute the number of pieces of each type of wood needed for the order.
4b. Next week she has one order for 6 plates, 2 large bowls, and 4 small bowls to ship out of town, and a second order for 12 plates and 9 small bowls for a local market. Set up and solve a matrix multiplication to compute the number of pieces of each type of wood needed to complete each order separately.

5. Four local restaurants offer all soups and sandwiches for a lunch special given in the table. On
Monday, you order 5 soups and 4 sandwiches. On
Wednesday, you order 2 soups and 7 sandwiches.
On Friday, you order 6 soups and 10 sandwiches.
5a. Set up and solve a matrix multiplication to find the cost from each restaurant per day of the week.
Restaurant #1
Restaurant #2
Restaurant #3
Restaurant #4
Soup
$2.75
$3.15
$2.90
$2.50
Sandwich
$7.20
$6.75
$7.00
$7.50
5b. Which restaurant has the best deal per day of the week?
5c. If you can only order from one of four restaurants for the entire week, which restaurant should you choose?
6. Three music classes at Central High are selling candy as a fundraiser. The number of each kind of candy sold by each of the three classes is shown in the following table.
Jazz Band Symphonic Band Orchestra
Almond Bars
Chocolate Chews
Mint Patties
Sour balls
300
240
150
175
220
330
200
150
250
400
180
160
The profit for each type of candy is sour balls, 30 cents; chocolate chews, 50 cents; almond bars, 25 cents; and mint patties, 35 cents. Use matrix multiplication to compute the profit made by each class on its candy sales.
7. The students at Central High are planning to hire a band for the prom. Their choices are bands A, B, and C.
They survey the Sophomore, Junior, and
Senior classes and find the following percentages of students (regardless of sex)
The student population by class and sex is: prefer the bands,
10 th
A
B
C
10 th




20
30
50
%
%
%
11 th
35 %
30 %
35 %
12 th
40 %
25 %
35 %




11
12 th th
Use matrix multiplication to find:
7a. The number of males and females who prefer each band.
Male


235


205
175
Female
225


215
190


7b. The total number of students who prefer each band.

8. The characteristics of a female reptile are shown in the following table. Suppose the initial female population for the heard is given by P
0


18 24 30 22 12 10

8a. What is the expected lifespan of this reptile?
8b. Construct the Leslie matrix for this population.
8c. Find the new population distribution after 15 years? (Hint: How many cycles or transitions ?)
8d. What is the total population of P
6
?
Age Groups
(yrs)
0 – 5
5 – 10
10 – 15
15 – 20
20 – 25
25 – 30
Birth
Rate
0
0.6
1.2
1.3
0.4
0
Survival
Rate
0.4
0.7
0.8
0.9
0.6
0
A
9. Suppose that a three-sector economy has this consumption matrix:
B
C
A




0
0
0
.
.
1
1
.
2
B
0 .
2
0 .
3
0 .
1
C
0
0
0 .
.
.
3
2
2




9a. A production matrix, P , follows, Find the internal consumption matrix product CP .
Find the external demand matrix D , where D

P

CP .
P





8
12 



 15 
9b. An external demand matrix, D , follows. Find the production matrix P for this economy. Recall that P


I

C
 
1
D .
D





6
8
12




10. A manufacturing company has divisions in Massachusetts, Nebraska and California.
The company divisions use goods and services from each other as shown in the following consumption matrix C.
C

MA
NE
CA
M A




0
0
0 .
.
4
.
3
1
NE
0 .
2
0 .
1
0 .
2
C A
0
0
0 .
.
.
3
5
4




10a. Find the total production needed to meet a final consumer demand of $50,000 form
Massachusetts, $30,000 from Nebraska, and $40,000 from California. (Recall that
P


I

C
 
1
D ).
10b. What will the internal consumption ( CP ) be for each division to meet the demands in part a?