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Why is it important that the matrix is invertible? Give and example.

Why is it important that the matrix is invertible? Give and example.

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.2: Arithmetic Sequences And Partial Sums
Problem 73E: Falling Object An object with negligible air resistance is dropped from the top of the Willis Tower...
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Hill Cypher At the beginning of the 1900 the Hill Cypher was developed to communicate secret messages. The letters of the alphabet are represented by 0 through 25. So for example the number 2 corresponds to the letter C. The Hill Cypher works in this way: Bob wants to communicate a secrete message to Alice. Bob will chose a secret word such as TANGERINE. The word will be translate into a sequence of numbers 19 0 13 17 To encode the message 19 0 13 6 4 17 9 13 4, which we can display as a 3 x 3 matrix 9. 6. 4 13 4 RHI, Bob first translates it into a vector (17 7 8), then computes the vector 19 13 17 11 6. 4 17 9. 13 4 8 25 which translate to the encoded message LGZ, which is the message that Alice receives. Because we have chosen a 3 x 3 matrix, if we have to communicate a longer message such as RHINOCEROS, we will need to divide the word in bits of 3 letters, adding at the end some extra (mute) letters. In this project you will explore some details that need to be worked out. 1. Compute the encoded message for R.HINOCEROS. 2. Consider the sentence linear algebra is really fun and translate in a sequence of numbers. 3. Encode the message using the word TANGERINE. 4. When the receiver recives your message, how are they recovering the original message? 5. Try encoding the message with the secret word RASPBERRY. 6. Can you find the inverse of the matrix coming from TANGERINE? What about the matrix com- ing from RASPBERRY? What about the matrix that come from the secret word PERSIMMON? (remember that you are using modulo 26 algebra) 7. Why is it important that the matrix is invertible? Give and example. 8. If Eve wants to decode the message, she will need to know the inverse of the matrix you used to decode the message. How many 3 x 3 matrices are there with entries that belong to the set {0, 1, 2, 3, ..., 25}?
Transcribed Image Text:Hill Cypher At the beginning of the 1900 the Hill Cypher was developed to communicate secret messages. The letters of the alphabet are represented by 0 through 25. So for example the number 2 corresponds to the letter C. The Hill Cypher works in this way: Bob wants to communicate a secrete message to Alice. Bob will chose a secret word such as TANGERINE. The word will be translate into a sequence of numbers 19 0 13 17 To encode the message 19 0 13 6 4 17 9 13 4, which we can display as a 3 x 3 matrix 9. 6. 4 13 4 RHI, Bob first translates it into a vector (17 7 8), then computes the vector 19 13 17 11 6. 4 17 9. 13 4 8 25 which translate to the encoded message LGZ, which is the message that Alice receives. Because we have chosen a 3 x 3 matrix, if we have to communicate a longer message such as RHINOCEROS, we will need to divide the word in bits of 3 letters, adding at the end some extra (mute) letters. In this project you will explore some details that need to be worked out. 1. Compute the encoded message for R.HINOCEROS. 2. Consider the sentence linear algebra is really fun and translate in a sequence of numbers. 3. Encode the message using the word TANGERINE. 4. When the receiver recives your message, how are they recovering the original message? 5. Try encoding the message with the secret word RASPBERRY. 6. Can you find the inverse of the matrix coming from TANGERINE? What about the matrix com- ing from RASPBERRY? What about the matrix that come from the secret word PERSIMMON? (remember that you are using modulo 26 algebra) 7. Why is it important that the matrix is invertible? Give and example. 8. If Eve wants to decode the message, she will need to know the inverse of the matrix you used to decode the message. How many 3 x 3 matrices are there with entries that belong to the set {0, 1, 2, 3, ..., 25}?
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