Sec 8.1 Matrices Review Voting methods & Matrix Applications

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2 Sec 8.1 Matrices Review Voting methods & Matrix Applications Name: Use the following matrices for the problems Show ALL WORK by HAND and check with the calculator A B 7 5 C 5 x D E 0 7 y What are the dimensions of Matrix D? T D 3. C E=. C D = 5. 2B 3A = 6. D C = 7. B D= 8. B E =

3 Use the following matrices for the problems Show ALL WORK by HAND and check with the calculator A B C D x E y 9. C 22 + D 31 B 21 = A 11. T D D 12. Create a 2x2, 3x3, and 5x 5 IDENTITY matrices 13. Are Matrices A and B inverses? Demonstrate how you can tell.

4 1. Determine possible values for a and b that would make the following matrix statement true. 2 2 a b 1. Determine possible values for a, b, and c that would make the following matrix statement true. 1 a b 8 c a = a = b = The determinant of 5 3 b = by hand and show work. 1. Find the determinant of c = by hand and show work Find the inverse of by hand and show work. 16. Find the inverse of by hand and show work. 17. On one weekend, the Goxfield Theater reported the following ticket sales for three first-run movies, as shown in the matrix at the right. If the ticket prices were $6 for each adult and $ for each child, what were the weekend sales for each movie?

5 Sec 8.2 Informational Matrices Voting methods & Matrix Applications Name: 1. A flu epidemic occurs in Middletown schools and each student is either susceptible, sick, or infected. The percentage of students in each category by grade The population distribution of the Middletown school district is level is given below. given below Girls Boys S Elementary Middle High School School School Susceptible 60 % 25% 15% Sick P 65% 15% 20% Infected 70% 10% 20% a. Describe the value of S 3 2 and what it represents. Elementary 1610 MiddleSchool 830 HighSchool b. Describe the value of P 2 2 and what it represents. c. Find S P. (You can use your calculator if you would like). INCLUDE LABELS d. Describe the value of P 2 1 S and what it represents. e. Based on the flu epidemic in Middletown schools, how many sick girls are there? f. Based on the flu epidemic in Middletown schools, how many boys are infected? 2. A study is being conducted by Time Warner about the network spending of its Network HBO and 2 online competitors. The percentage of spending categories by each network is shown below. S NetFlix HULU HBO Movies 60% 35% 70% Mini Series 30% 30% 30% Current TV 10% 35% 0% a. Describe the value of S 2 3 and what it represents. The amount of spending per month by each company is shown below. Monthly Spending P NetFlix $85 million HULU $33 million HBO $912 million b. Describe the value of P 31 and what it represents. c. Find S P. (You can use your calculator if you would like). INCLUDE LABELS d. Describe the value of 11 S P and what it represents. e. Based on the study, how much do the 3 companies spend on Mini Series? f. Based on the study how much does HBO spend on Movies and their licensing?

6 3. following matrices show information collected by an apparel store in Magic Kingdom park at Disney World The matrix [N] shows the number of items sold on Monday N Infant / Toddler Youth Adult Jackets Shirts Hats The matrix [P] shows the Sale Price and Profit of each type of item. Sale Profit Price P Jackets Shirts Hats $75 $0 $30 $15 $16 $9 a. Describe the value of N 23 and what it represents. b. Describe the value of P 2 1 and what it represents. c. Find N P. (You can use your calculator if you would like). INCLUDE LABELS d. Describe the value of P 3 2 N and what it represents. e. Based on the information what was the total Sales on Monday for Adult Apparel? f. Based on the information what was the total Profit made on Monday on Infant Apparel?

7 Sec 8.3 Basic Voting Methods Voting methods & Matrix Applications Name: Consider the following preference schedules for an election. Droid (Motorola) iphone 6 (Apple) iphone 6 (Apple) Galaxy 5 (Samsung) Galaxy 5 (Samsung) Bold Z30 (Blackberry) Galaxy 5 (Samsung) iphone 6 (Apple) Bold Z30 (Blackberry) Bold Z30 (Blackberry) iphone 6 (Apple) Galaxy 5 (Samsung) Bold Z30 (Blackberry) Droid (Motorola) Droid (Motorola) Droid (Motorola) How many preference schedules are possible (if ties are not permitted)? #1) 2. Who is the plurality winner? What is the percentage of 1 st place votes each received? 3. How many first place votes would be needed in this example for there to be a majority winner? If there is a majority winner who is it?. Who is the run off winner? #2) #3) #3) #) 5. Who is the sequential run off winner? #5) Droid (Motorola) iphone 6 (Apple) iphone 6 (Apple) Galaxy 5 (Samsung) Bold Z30 (Blackberry) Galaxy 5 (Samsung) Bold Z30 (Blackberry) Droid (Motorola) Droid (Motorola) Droid (Motorola) What is each candidates Borda count? Galaxy 5 (Samsung) iphone 6 (Apple) Bold Z30 (Blackberry) Bold Z30 (Blackberry) iphone 6 (Apple) Galaxy 5 (Samsung) #6) Droid: ip6: Glxy5: Who is the Borda Count winner? ***Demonstrate how this can be done with matrix multiplication**** 7. What is each candidates Condorcet winner? #7) Drd ip6 G5 Z30 Drd * ip6 * G5 * Z30 * Z30: #6) The BIG QUESTION: WHO REALLY WINS? SUPPORT YOUR REASONING.

8 Twenty-two Discrete Math students are arguing over which fast food restaurants and listed their preferences below. Five Guys Wendy s Zaxby s Chic-fil-a Chic-fil-a Chic-fil-a Chic-fil-a Wendy s Zaxby s Zaxby s Wendy s Zaxby s Wendy s Five Guys Five Guys Five Guys In your own words, give a description of the plurality winner What percentage of 1 st place votes does each of the following choices have? Choice Five Guys Chic-fil-a Zaxby s Wendy s Percentage of 1 st place votes Who is the plurality winner? 2. a.what is the minimum number of first place votes needed in this example for there to be a majority winner? b.if there is a majority winner who is it? 3. In your own words, give a description of the run off winner : 2) #3a) #3b) #) Who is the run off winner?. In your own words, give a description of the sequential run off winner : 5) Who is the sequential run off winner? 5. In your own words, give a description of the Borda Count winner (on a separate page show how this might be done using Materices): a. Give the Borda Count for each letter: b. Who is the Borda Count winner? 6. Determine the Condorcet Winner. #6a) 5G: C: Zx: W: 5G C Zx W 5G * C * Zx * W * #6b)

9 Sec 8. Matrices Transformations Voting methods & Matrix Applications Start by creating a picture on a Cartesian coordinate system (preferably in the first quadrant) The picture at the right would be represented by the matrix: Enter this into the Matrix [A] in the calculator. Press MATRX,, ENTER. Change the dimensions of the matrix to match the points of your picture. For our example we will need to change the dimensions to 2 x 7 Name: To return to the home screen press 2 nd MODE To have the calculator show your original picture. Press MATRX,, 8. This will bring up Matr list( on the calculator. This function will enable us to put our matrix into the table of the calculator which can be graphed. Next, press MATRX, 1. Before the Matrix can be transformed into the table it has to be turned vertically or TRANSPOSED. Press MATRX,, 2,,, 2 nd, 1,,, 2 nd, 2, ), ENTER. Next hit 2 nd, Y= (Stat Plot), 1. Make sure your screen has the following options highlighed Finally push ZOOM, 6. Dilations To make the object dilate, using (0,0) as the center of dilation, multiply the matrix by a scalar. To have the calculator show your original picture. Press MATRX,, 8. This will bring up Matr list( on the calculator. This function will enable us to put our matrix into the table of the calculator which can be graphed. Next, press, 2, MATRX, 1. Before the Matrix can be transformed into the table it has to be turned vertically or TRANSPOSED. Press MATRX,, 2,,, 2 nd, 3,,, 2 nd,, ), ENTER. Next hit 2 nd, Y= (Stat Plot), 2. Make sure your screen has the following options highlighed Finally push ZOOM, 6. Translation To translate an object we have to set up a translation matrix. We can enter this in matrix [B] Shift left 7 and down Rotation To rotate an object we have to set up a rotation matrix. We can enter this in matrix [C] Rotation by 90

10 General Rotation To rotate an object we have to set up a rotation matrix. We can enter this in matrix [C] (be sure you are in DEGREE mode.) cos160 sin 160 sin 160 cos160 Rotation by 160 Press the MODE key and switch to DEGREE mode Combination Transformations (Rotate 200 and Translate Right 6) To translate an object we have to set up the rotation matrix. We can enter this in matrix [C] (be sure you are in DEGREE mode.) and a translation matrix in [B]. B Shift right 5 cos 200 sin 200 C sin 200 cos 200 Press the MODE key and switch to DEGREE mode Rotation by 200 Programmers often use matrices to write visual code even for the mouse cursor For example let s suggest that the points A,B,C,D,E,F,G represent a mouse cursor. Programmers would use the matrix: a. Draw the original mouse cursor on the graph using a blue colored pencil. S A 3 2 B 3 C 5 2 D 5 5 E 2 5 F 3 G 2 3 b. Usually mouse cursors are translated. To do this programmers add by a translation matrix. Add the following translation matrix to [S] and draw the new image using a red pen. S c. To rotate an object counter clockwise programmers multiply a rotation matrix and [S]. Multiply the following and graph using a black or gray pencil. (should be degree mode) 130 sin130 cos sin 130 cos130 S

11 A Matrices are used to describe most graphics on computer (including B computer/video games). Consider the following Phoenix bird. As a matrix it would be described as: A B C D E F G H I J A Once entered as a matrix several graphical transformations can be performed using matrix operations. DILATIONS: To dilate (bigger or smaller) from the origin point, you would only need to multiply the original matrix by a scalar multiple. e. g would make the picture twice as big would make the picture shrink the picture to half size TRANSLATION Matrices: h k In the above example all you would need to do is change h to the a number to translate the picture left or right OR change k to a number to tran the picture up or down. ROTATION Matrices: cos sin sin cos In the above example all you would need to do is change theta to an angle you wish to rotate the shape about the origin. ROTATION Matrices: cos sin sin cos In the above example all you would need to do is change theta to an angle you wish to rotate the shape about the origin.

12 Start by creating a picture on a Cartesian coordinate system (preferably in the first quadrant) The picture at the right would be represented by the matrix: Enter this into the Matrix [A] in the calculator. Press MATRX,, ENTER. Change the dimensions of the matrix to match the points of your picture. For our example we will need to change the dimensions to 2 x 26 To return to the home screen press 2 nd MODE To have the calculator show your original picture. Press MATRX,, 8. This will bring up Matr list( on the calculator. This function will enable us to put our matrix into the table of the calculator which can be graphed. Next, press MATRX, 1. Before the Matrix can be transformed into the table it has to be turned vertically or TRANSPOSED. Press MATRX,, 2,,, 2 nd, 1,,, 2 nd, 2, ), ENTER. Next hit 2 nd, Y= (Stat Plot), 1. Make sure your screen has the following options highlighed Finally push ZOOM, 6. You can turn the axes on or off by pressing 2 nd, ZOOM and selecting the option AxesOff. Dilations To make the object dilate, using (0,0) as the center of dilation, multiply the matrix by a scalar. To have the calculator show your original picture. Press MATRX,, 8. This will bring up Matr list( on the calculator. This function will enable us to put our matrix into the table of the calculator which can be graphed. Next, press, 2, MATRX, 1. Before the Matrix can be transformed into the table it has to be turned vertically or TRANSPOSED. Press MATRX,, 2,,, 2 nd, 1,,, 2 nd, 2, ), ENTER. Finally push ZOOM, 6.

13 If each screen was viewed in rapid succession it would give the appearance of flying into the picture. Create your own version of a drawing that you would like to animate using transformational matrices Attempt #1 Attempt #2 Using up to 10 frame of animation, show rough sketches of your anticipated slides to create your still animation. Slide # 1 Slide # 2 Slide # 3 Slide # Slide # 5 Slide # 6 Slide # 7 Slide # 8 Slide # 9 Slide # 10

14 Summary: With a couple of similar pictures we can make an animation of a fish that blows bubbles, an eye that s winking, or just about anything you wish to make move. Animations for cartoons, movies, and flip books are created by making several similar pictures each with a slight change showing where the object has moved too. Creating a fish that moves to the right would require creating the following pictures: If these pictures were played in rapid succession on the TI-83 the fish would appear to move to the right. As you create each picture on the TI-83 you will need to store the picture. If these pictures were played in rapid succession on the TI-83 the heart would appear to beat. As you create each picture on the TI-83 you will need to store the picture. Storing your Pictures: First display the picture you would like to store as your first picture in your animation. You can turn on a plot by selecting it from the Y= button. o o Press Y =. Highlight Plot1 by pressing the cursor up. Pressing ENTER, will either turn on or turn off the plot (if the plot is highlighted then it is turned on). For the Heart example turn Plot1 on and Plot2 off. Press GRAPH DRAW With the picture you wish to store on the graph screen, press: 2 nd, PRGM,, 1. The calculator should say StorePic. The TI-82/83/8 allows you to save up to 10 different pictures. Next, we will need to select where we would like to store the picture. Press: VARS,. Now select where you would like to store the picture in Pic1 through Pic9. Select Pic1 if this is the first picture. After selecting the appropriate place to store the picture press ENTER. After pressing enter the picture that you are storing should re-appear.

15 Next display the picture you would like to store as your second picture in your animation. o Press Y =. Highlight Plot2 by pressing the cursor up. Pressing ENTER, will either turn on or turn off the plot (if the plot is highlighted then it is turned on). For the Heart example turn Plot1 off and Plot2 on. o Press GRAPH DRAW With the picture you wish to store on the graph screen, press: 2 nd, PRGM,, 1. The calculator should say StorePic. The TI-82/83/8 allows you to save up to 10 different pictures. Next, we will need to select where we would like to store the picture. Press: VARS,. Now select where you would like to store the picture in Pic1 through Pic9. After selecting the appropriate place to store the picture press ENTER. After pressing enter the picture that you are storing should re-appear. Repeat this process until all of the pictures are stored. Next, we will need to create a program that displays each of the pictures in rapid succession to give the illusion of movement. It may help at this point to turn off all of the plots under Y =. Creating an Animation Program: Start by creating a new program to do this press: PRGM and select NEW by pressing the,, ENTER and then typing in a name such as A, N, I, M ENTER. (FnOff) Now, we should have an almost blank screen ready to be programmed. The first thing we will need to do is force the calculator to be set up correctly. The first line we will put in a command to turn off any graphed equations. Press VARS,,,and 2 ENTER. (LBL 1) Next, we will need to set up a label so that the program can loop continuously through the animated sequence. Press: PRGM, 9, 1, ENTER. (ClrDraw) Now, we will need to clear the current screen to begin the animation. Press: 2 nd, PRGM, 1, ENTER. (RecallPic Pic1) Then, we will need to recall the first picture in the animation sequence. Press: 2 nd, PRGM,, 2, VARS,, 1, ENTER. Example (For) If we were to immediately clear this picture and display the next picture the animation would take place too quickly. So, we will need to set up some type of delay while this picture is being displayed. This can be done with a quick FOR Loop as shown in example at the right The FOR command is found by pressing PRGM,, X,T,,n,,,1,,,3,0,,,1, ), ENTER. The (X, 1, 30, 1) shown in Example 2 stands for (Variable, Beginning Count Number, Ending Count Number, Count By). PRGM, 7 (End) ENTER. Try changing the Ending Number (30) for different delays. (ClrDraw) Now, we will need to clear the first picture. Press: 2 nd, PRGM, 1, ENTER. (RecallPic Pic2) Then, we will need to recall the first picture in the animation sequence. Press: 2 nd, PRGM,, 2, VARS,, 2, ENTER. Again, if we were to immediately clear this picture & display the next the animation would take place too quickly. We need to create another FOR-Loop. PRGM,, X,T,,n,,,1,,,3,0,,,1, ), ENTER, PRGM, 7,ENTER. If there are additional pictures to include in the animation the procedure is the same for adding more pictures. (Goto 1) Finally, we need to loop the animation back to the beginning to set up a continuous animation sequence. Press: PRGM, 0, 1. Go back to the HOME SCREEN 2 nd, MODE and Execute PRGM the ANIM program. Press the ON button to interrupt the animation.

16 PROEJCT SHEET (INITIAL PICTURES) NAME:

17 PROEJCT SHEET (TRANSFORMATIONS) NAME:

18 PROEJCT SHEET (TRANSFORMATIONS) NAME:

19 PROEJCT SHEET (TRANSFORMATIONS) NAME:

20 PROEJCT SHEET (TRANSFORMATIONS) NAME:

21 PROEJCT SHEET (TRANSFORMATIONS) NAME:

22 PROEJCT SHEET (TRANSFORMATIONS) NAME:

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24 Sec 8.5 Population & Leslie Matrices Voting methods & Matrix Applications Name: Suppose a new species of pig Sus-Gigantis in a small region had the following population Age Groups Birth Rate Survival Rate Age Groups (in years) Year: # of Pigs in initial population a. Calculate how many 0-2 year olds there will be in the next 5 year cycle. b. Complete the Table for the next cycle and show your steps. Age Groups (in years) in initial pop (Year:2008) Population (Year: 2010) c. Complete the Table for the next cycle and show your steps. in initial pop (Year:2010) Age Groups z (in years) Population (Year: 2012)

25 DO THIS PROBLEM AGAIN BUT USING MATRICES. Suppose a species of pig, Sus-Gigantis, has characteristics as described below: Age Groups Birth Rate Survival Rate Age Groups (in years) Year: # of Pigs in initial population a. Create the initial population matrix and the corresponding LESLIE matrix. b. How many Leslie Cycles would have been completed by the year 2010? c. Using your calculator and the Matrices created in part a of this question determine the population of monkeys for each age group in the year Population (Year: 2010) Age Groups (in years) d. How would you suggest estimating how many pigs were in each age group in the year 2011? e. What would be your estimate for the year 2011 using your suggestion from part d? Population (Year: 2011) Age Groups (in years) f. Why is it necessary for the survival rate of the last group to be 0 for this model to work?

26 2. Suppose a certain species of Elephants has characteristics as described: Age Groups Birth Rate Survival Rate Age Groups (in years) Year: # of elephants in initial population a. Calculate how many 0-5 year olds there will be in the next 5 year cycle. b. Complete the Table for the next cycle and show your steps. Age Groups (in years) in initial pop (Year:2000) Population (Year: 2005) c. Create the initial population matrix and the corresponding LESLIE matrix. d. How many Leslie Cycles would have been completed by the year 2025? e. Using your calculator and the Matrices created in part a of this question determine the population of elephants for each age group in the year Age Groups (in years) Population (Year: 2025) f. It appears that the overall population is declining. If this trend continues how long (or in what year) would you estimate until the population of all of the subgroup categories finally fall below 1 elephant.

27 g. BONUS: How would you suggest estimating elephant populations in each age group in 2026? h. What would be your estimate for the year 2026 using your suggestion from part e? Population (Year: 2026) Age Groups (in years)

28 Sec 8.6 Markov Chains Voting methods & Matrix Applications Name: A Markov chain is a process that arises naturally in problems that involve a finite number of events or states that change over time. A company is doing a research study in the metro-atlanta area about smart phones (either android or iphone) and determined that 56% of the sample owned iphones and the other % owned Android based phones. After following the customers for a while, the researcher determined that each year after that 80% of the iphone owners purchased a new iphone again when it came time to renew their contract. The remaining 20% switched to Android. The researcher also determined that 65% of android owners purchased a new Android phone again when it came time to renew their contract. The remaining 35% of android owners switched to an iphone. a) Create a tree diagram showing the percentages of each owner after one cycle of renewals. b) Create the initial distribution matrix D o c) Create a Transition Matrix. d) In the study there were initially 22 iphone owners and 176 Android owners. If each participant purchases a new phone every 2 years, determine how many of the participants would own iphones and how many would own Androids 8 years after the study began? Suppose that in the Northwest region of the country, Ford dealers make up 0% of the automobile sales, GM dealers make up 5% and foreign car dealers 15%. The foreign car dealers decide to offer a nointerest-for-6-months-after-purchase incentive plan and during the next year of business, the following transition matrix evolves: FROM Ford GM Foreign Ford GM Foreign Based on this matrix, what percentage of GM drivers will switch to Ford each year? TO After year 3, what percentage of drivers will be driving Fords? What percentage of drivers will be driving Fords long-term?

29 A food service director for a local high school conducted a survey in hopes of predicting the number of students who will eat in the cafeteria in the future. The results of the survey are as follows: If a student eats in the cafeteria on a given day, the probability that he or she will eat there again the next day is 70% and the probability that he or she will not eat there is 30%. If a student does not eat in the cafeteria on a given day, the probability that he or she will eat in the cafeteria the next day is 0% and the probability that he or she will not eat there is 60%. On Monday, 70% ate at the cafeteria and 30% at somewhere else a) Create a tree diagram showing the percentages of each owner after one cycle of renewals. b) Create the initial distribution matrix D o c) Create a Transition Matrix. d) Find D 3 e) How many students could be expected to eat at the cafeteria on Friday if there were 350 students that ate at the cafeteria and 150 that did not on Monday?