Discrete Math Through Applications (DMTA) Lesson 1
MATRIX REVIEW
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 / Sal’sPizza / $12.16 / $10.10 / $10.86 / $10.65
Drink / $1.15 / $1.09 / $0.89 / $1.05
Salad / $4.05 / $3.69 / $3.89 / $3.85
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 A14or A1,4.
1. What is the value of element:A21? ______A12? ______A34? ______
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 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?
matrix: _____ matrix : ______matrix : ______ 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 column matrix.
Write the column matrix for Vinny’s.
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 row matrix.
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, 55pairs 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
MATRIX OPERATIONS REVIEW
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 is SYMMETRIC if Sij = Sji where i and j = 1, 2, 3,…, n. (Matrix is a reflection across the diagonal)
- Is matrix Kormatrix Lsymmetric?
- The elements and values of the main diagonal for matrix K are K11 = 0, K22 = –1, and K33 = 4. Identify the elements and the values of the main diagonal in matrix L.
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.
Examples: Identify which operations have a solution or not.
a. b. c. d.
The operations of addition and subtraction are by corresponding elements of the matrices:
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.
AdditionExample:
SubtractionExample:
SCALAR MULTIPLICATION: In general, scalar multiplication is a process in which we multiply each element in a matrix A by a constant number k andk is referred to as the scalar. The resulting matrix is kA.
Scalar Multiplication Example:
MATRIX CALCULATOR COMMANDS:
STEP 1:Making a matrix
- Create the matrix: [2ND] [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: “[2nd] [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: [2ND] [Matrix] NAMESScroll to the matrix [ENTER]
MATRIX OPERATION PRACTICE:Try by hand, and then check with calculator
1)
a) Find A – B
b) Find 4C
c) Find C + A
d) Find 2A + B
2) , , and
a) Find L + N – M
b) Find L + 2N
c) Find N– 2M
d) Find 3L + 2M
e) Find 2(L + M)
f) 2M – 3N + 5L
3) Find the values of x, y, and z for
4) Find the values of a, b, and cfor
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 / FWeek 1 / 45 / 60 / 50 / 23 / 15
Week 2 / 25 / 30 / 35 / 42 / 47
Jessica
/ M / T / W / T / FWeek 1 / 32 / 35 / 33 / 35 / 36
Week 2 / 47 / 42 / 40 / 39 / 31
Discrete Math Through Applications (DMTA) Lesson 3
MATRIX MULTIPLICATION
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.
CASE #1: Row Matrix x Column Matrix
In general, if A is a row matrix and B is a column matrix, thenis a 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 row matrix Q (quantity matrix).
The prices of each snack can be displayed in a column matrix P (price matrix).
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.
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
CASE #2: Row Matrix x Multi-dimensional Matrix
IfA is a row matrix and B is a multi-dimensional matrix, thenis a 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.
To find each restaurant’s total cost of order by multiplying the row matrix A and matrix Tand
1. Write an interpretation of AT12.
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: , P: defined? ______dimension of QP______
b.Q: , P: defined? ______dimension of QP ______
c.Q: , P: defined? ______dimension of QP ______
d.Q: , P: defined? ______dimension of QP ______
MA / NE / CALoans / 230 / 440 / 680
Bonds / 780 / 860 / 940
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 to compute the given expressions.
a.3A
b.BA
c.BC
CASE #3: Multi-dimensional Matrix x Multi-dimensional Matrix
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 Amatrix 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 row matrix and B is a multi-dimensional matrix, thenis a row matrix.
Example:Recall the matrix T and suppose for your pizza party, you are considering three different options for pizza and salad combinationshown in the matrix B.
The different combinations you are considering are. If we multiply matrix B times matrix T, the product will be a matrix that will represent each restaurant’s cost for each option.
1. In the product matrix, BT11 represents the cost of four pizzas and three salads at Vinny’s. Interpret BT23 and BT32.
2. Use , ,to compute the given expressions.
2a. AB
2b. BA
2c. BC
2d. CB
3. Matrix Multiplication Practice: If possible find the solution.
a.
b.
c.
d.
e.
f.
North / South / EastModel Trains / 10 / 8 / 12
Model Cars / 6 / 5 / 4
Model Planes / 3 / 2 / 2
Model Trucks / 4 / 3 / 2
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 find the total sales at each location.
4b. If the stores decided to sell trains for $45, cars for $40, planes for $65, and trucks for $25, would they have made more money?
5.The table gives the nutritional information for the menu items at a restaurant.
Calories / Fat (g) / Cholesterol (mg)Cheeseburger / 450 / 40 / 50
Special / 570 / 48 / 90
Baked Potato / 500 / 45 / 25
French Fries / 300 / 30 / 0
Shake / 400 / 22 / 50
5a. Rosa orders a special, fries, and a shake. Max has a cheeseburger, a baked potato, and a shake. Set up an appropriate matrix multiplication for their orders and the nutritional information to find the total amount of national information for their orders.
5b. Daryl orders 2 cheeseburgers, a shake, and fries. Jordan orders a special, cheeseburger, baked potato, and two shakes. Set up an appropriate matrix multiplication for their orders and the nutritional information to find the total amount of national information for their orders.
Discrete Math Through Applications (DMTA) Lesson 4
LESLIE MODEL PART 1
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.
Age (months) / Birth Rate / Survival Rate0-3 / 0 / 0.6
3-6 / 0.3 / 0.9
6-9 / 0.8 / 0.9
9-12 / 0.7 / 0.8
12-15 / 0.4 / 0.6
15-18 / 0 / 0
- 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 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.
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 / 15-18Number / 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) / # / Survival Rate / Rats to move up age group0-3 / 15 / 0.6
3-6 / 9 / 0.9
6-9 / 13 / 0.9
9-12 / 5 / 0.8
12-15 / 0 / 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 1st to 2nd transition?
2)How many rats were able to survive from their current age group to the next age group?
Age (months) / # / Survival Rate / Rats to move up age group0-3 / 0.6
3-6 / 0.9
6-9 / 0.9
9-12 / 0.8
12-15 / 0.6
15-18 / 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 (years) / Birth Rate / Survival Rate0 – 2 / 0 / 0.6
2 – 4 / 0.8 / 0.8
4 – 6 / 1.7 / 0.9
6 – 8 / 1.7 / 0.9
8 – 10 / 0.8 / 0.7
10 – 12 / 0.4 / 0
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.
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)
Discrete Math Through Applications (DMTA) Lesson 5
LESLIE MODEL PART 2
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 therows 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.
Example #1: Leslie Model Rat Example from Lesson 4
Age (months) / Birth Rate / Survival Rate0-3 / 0 / 0.6
3-6 / 0.3 / 0.9
6-9 / 0.8 / 0.9
9-12 / 0.7 / 0.8
12-15 / 0.4 / 0.6
15-18 / 0 / 0
Age (months) / 0-3 / 3-6 / 6-9 / 9-12 / 12-15 / 15-18
Number / 15 / 9 / 13 / 5 / 0 / 0
1a. Create a matrix multiplication P0L with a row matrix, P0, for the initial population distribution and the Leslie Matrix, L to find the population after one cycle. Compare to Lesson 4 answer.
1b. How might you calculate the next population distribution (2 cycles) using your Leslie Matrix?
In general if you know the initial population distribution, P0, 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 P25, P26, and P27and their total populations.
1e. Calculate the rate of growth (percent of change) of TOTAL populations between these cycles.
Age (years) / Birth Rate / Survival Rate0 – 2 / 0 / 0.6
2 – 4 / 0.8 / 0.8
4 – 6 / 1.7 / 0.9
6 – 8 / 1.7 / 0.9
8 – 10 / 0.8 / 0.7
10 – 12 / 0.4 / 0
Example #2: Leslie Model Deer Example from Lesson 4
Age / 0-2 / 2-4 / 4-6 / 6-8 / 8-10 / 10-12Deer / 50 / 30 / 24 / 24 / 12 / 8
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)
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 P0?
3c. Find P3.
3d. What is the total number of female bugs in P5?
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 monthsBirth 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 P25, P26, and P27.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
LEONTIEF INPUT-OUTPUT MODEL PART 1
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.