Solve the matrix equation AX B for X by using A.(1-3) Use the Inverse Matrix Calculator Link to check your work

Transcription

1 Name: Math 1324 Activity 9(4.6)(Due by Oct. 20) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please allow them to work the problems on their own! If you would like to help them with similar problems, here are the related homework problems: pg. 239:9-37 odd, odd.. Thanks! Dear Student, Don t even think about asking for help on the problems in this activity! The recommended homework problems have answers in the book, so do them first for practice!!! 1. 1 Solve the matrix equation AX B for X by using A.(1-3) Use the Inverse Matrix Calculator Link to check your work A, B A, B

2 A 0 1 2, B

3 4. Use the inverse of the coefficient matrix to solve the following systems.(4-5) Use the Inverse Matrix Calculator Link to check your work. x y 3z 3 2x 4y 4z 6 x y 4z 1

4 5. 2x 2y 4z 3 2x6y16 3x 3y 5z20

5 6. Matrix inverses can provide a simple and effective procedure for encoding and decoding messages. To begin, assign the numbers 1-26 to the letters of the alphabet, as shown below. Also assign the number 27 to a blank to provide for a space between words. (A more sophisticated code could include both upper and lower case letters and punctuation symbols.) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Blank So the message SECRET CODE corresponds to the sequence Any matrix whose entries are positive integers and whose inverse exists can be used as an encoding matrix. For example, to use the 2 2 matrix A to encode the above message, first we divide the numbers in the sequence into groups of 2 and use these groups as the columns of a matrix B with 2 rows: B (Notice that an extra blank was added at the end to make the columns come out even.) Then we multiply this matrix on the left by A: AB The coded message is This message can be decoded simply by putting it back into matrix form and multiplying on the 1 left by the decoding matrix A 1. Since A is easily determined if A is known, the encoding matrix A is the only key needed to decipher messages encoded in this manner. Although simple in concept, codes of this type can be very difficult to crack. 1 2 Given the encoding matrix A 1 3, a) Encode the message THE SUN ALSO RISES using matrix A.

6 b) Encode the message THE GRAPES OF WRATH using matrix A. c) The following message was encoded using matrix A. Decode this message: Use the Inverse Matrix Calculator Link to check your work d) The following message was encoded using matrix A. Decode this message: Use the Inverse Matrix Calculator Link to check your work A small telephone system connects three cities. There are four telephone lines between cities 3 and 2, three telephone lines connecting city 3 with city 1, and two telephone lines between cities 1 and 2. See the diagram. a) Write a matrix B to represent this information. City 1 City 2 city 1 city 2 city 3 city 1 city 2 city 3 B City 3

7 b) Find 2 B. c) How many lines that connect cities 1 and 2 go through exactly one other city? d) How many lines that connect cities 1 and 2 go through at most one other city? Bonus#1: There is an interesting five-digit number. With a digit of 1 placed on its right side, it is three times as large as with a digit of 1 placed on its left side. What is the number? Bonus#2: The following figure consists of six congruent squares, and it has an area of 294. Find the perimeter of the figure.

8 Bonus#3: B H A C H I F A N I N I