For this project, your work should be neatly done in a Microsoft Word document titled YOUR NAME Matrices Project.  When you have completed your project, save your Microsoft Word document as a PDF and then submit the file in your Google Drive by completing the following steps: 1. log into your Google Drive  2. click on "Shared With Me" at the left.  3. click on the folder labeled as your name.  4. click on the upload button that is directly to the right of the red create button.   5. select the pdf file and click upload. A video showing this process can be viewed by clicking on the following link: Uploading your project for grading.  You will be graded based on the following rubric: Matrices Rubric.

Part 1: Secret Messages in Pictures

Background

Terrorist groups sometimes hide messages in photographs in a process called steganography.  On a computer screen, a picture is made up of a series of thousands of dots and bytes, and not every single byte is needed to make the picture.  A coded message can easily be hidden in the unused parts of the image. 

How does this work?  Every bitmap picture viewed on a computer is made up of thousands of very small squares called pixels, as shown in the picture below.


The computer views these pixels as numbers, not colors.  For example, the pixilated image above could be translated into the values below:


Someone sending a secret message as a picture could use encryption through matrix multiplication. 

Assignment

As a member of the local police force, you are trying to take down a smuggling group that has been operating in the area.  A week ago, the S.W.A.T. team raided a house of one of the suspected members and found an encryption key (encoding matrix) that the suspected smuggling group uses to pass secret messages.  The encryption key is shown below:

   
Three encrypted computer images were just intercepted.  An informant tells you he overheard the messages are from the leader of the group and contain the following information:

Message 1: The day the shipment of smuggled goods will arrive.
Message 2: The weight of goods being smuggled in the upcoming shipment in thousands of pounds.
Message 3: The number of ship container the smuggled goods are in.

You have been asked to use your knowledge of cryptography to decode the message you were assigned by the police chief.  You can view your intercepted message by clicking on the following link: Intercepted Messages.
 
Once decoded, the values in the matrix represent colors that should go in each corresponding space.  The following table shows the color that corresponds with each number.


1. Decode the intercepted message.  Show all work in your project document.  You can use a table to represent all matrices.

2. Use the Blank Google Drawing in the folder labeled as your name in the Shared with Me section in your Google Drive to reconstruct the image based on the color chart above.  Click on the cell you wish to color, click on the paint bucket and change the color to the color you want to color the cell (Hint: the colors above are in the same order as the second row of colors in the paint bucket). Include a copy of your reconstructed image in your project document.

3. In a complete sentence, describe what information was in your message, as well as the answer the intercepted message contained.

4. The Mayor is amazed at your work and has asked you to tell him how you did it.  Describe, in English and in complete sentences, how you determined the information from your intercepted message.  Make sure that your description is worthy of the Mayor's eyes to receive full credit.

Part 2: Truck Rentals 
 
Lisa Crawford is getting into the moving-truck rental business in three nearby counties.  She has the funds to buy about 100 trucks.  Her studies show that 20% of the trucks rented in Durham County go to Wake County, 15% go to Johnston County, and the rest stay in Durham County.  From Wake County, 25% of rentals go to Durham County, and 55% stay in Wake County, and the rest move to Johnston County.  From Johnston County, 40% of rentals go to Durham County, 30% go to Wake County, and the rest stay in Johnston County.
 
a) Draw a transition diagram that displays this information.  (You can do this using Google Drawing).
 
b) Write a transition matrix that represents this scenario.  List your rows and columns in the order Durham, Wake, Johnston.  You can use a table to represent your matrix in your document
 
c) What is the sum of the entries in row 1?  Row 2? Row 3?  Why does this sum make sense?
 
d) If Lisa starts with 45 trucks in Durham County, 30 trucks in Wake County, and 25 trucks in Johnston County, and all the trucks are rented one Saturday, how many trucks will she expect to be in each county the next morning?