Question

Recall that in order to start the scavenger hunt, Darnell needs to send an encoded message to Raya.a) Based on the table, what is the number string that corresponds to the text “Hello Raya”? (1 point) b) How many 2 x 1 matrices will be in the sequence? How do you know that? (2 points) c) Write the sequence of 2 x 1 matrices that Darnell will use to encode. (1 point) d) What is the enumerated matrix B? (2 points) e) What is A × B? (Show your work) (4 points) AssignmentPart 1: Use an encoding matrix.Raya set up a short scavenger hunt using matrices to encode the location of a wooden token. The scavenger hunt might involve several locales on the school campus, such as the auditorium, cafeteria, gymnasium, library, and the quad (common area where students gather). To receive the first clue, Darnell has to first encode a message to Raya.To encode the message, Darnell was given a 2 x 2 encoding matrix, A. Each letter of the alphabet is assigned a numerical value, and the value ‘0’ represents a space and is indicated in the code by an underscore ( _ ).0_ 9I18R1A10J19S2B11K20T3C12L21U4D13M22V5E14N23W6F15O24X7G16P25Y8H17Q26Z The method in which the messages are encoded is as follows.“Test one” coordinates to the number string: 20 5 19 20 0 15 14 5 TEST_ ONE2051920015145 The encoding matrix is 2 x 2, so the enumerated message above is broken up into a sequence of 2 x 1 matrices, then combined to create matrix M. The encoded matrix is the result of matrix multiplication between the coded matrix, A, and the enumerated message, M. Recall that in order to start the scavenger hunt, Darnell needs to send an encoded message to Raya.a) Based on the table, what is the number string that corresponds to the text “Hello Raya”? (1 point) b) How many 2 x 1 matrices will be in the sequence? How do you know that? (2 points) c) Write the sequence of 2 x 1 matrices that Darnell will use to encode. (1 point) d) What is the enumerated matrix B? (2 points) e) What is A × B? (Show your work) (4 points) Part 2: Use a decoding matrix.Raya sent an encoded message back to Darnell that provides the location of the next clue.To decode the message, Darnell will need to use the inverse of the 2 x 2 encoding matrix, A. Complete the following problems to determine the next clue.a) What is the determinant of A? Show your work or explain. (3 points) b) What is the inverse matrix of A? Show how to get A–1. (4 points) c) What is A–1 × C? (4 points) d) Describe how to use the resultant matrix from part c to decode the message. What does the decoded message say? (4 points) Part 3: Find the location of the wooden token.Darnell then found a 3 x 3 encoding matrix, D, and must now use it to encrypt the word “bleachers” so that he will receive the matrix containing the location of the wooden token.a) What is the enumerated matrix E that corresponds to the text “bleachers”? (3 points) b) What is D × E? (4 points) The resultant matrix containing the location of the wooden token is the following: c) What is the determinant of D? Show your work or explain. (4 points) d) What is the inverse matrix of D? Show how to get D–1. (5 points) e) What is D–1 × F? (5 points) f) Based on the resultant matrix, what is the number string that corresponds to the decoded message? (2 points) g) What does the decoded message say? (2 points)

Answer 1

PART ONE

Answers:

a) 8, 5 12, 12, 15, 0, 18, 1, 25, 1

b) The sequence will have 5 2x1 matrices.

c)The 2x1 matrices are:

`\begin{bmatrix}{8} & {} \\ {5} & {}\end{bmatrix}\begin{bmatrix}{12} & {} \\ {12} & {}\end{bmatrix}\begin{bmatrix}{15} & {} \\ {0} & {}\end{bmatrix}\begin{bmatrix}{18} & {} \\ {1} & {}\end{bmatrix}\begin{bmatrix}{25} & {} \\ {1} & {}\end{bmatrix}`

d)

`\begin{gathered} 8\text{ }12\text{ }15\text{ }18\text{ }25 \\ 5\text{ }12\text{ }0\text{ }1\text{ }25 \end{gathered}`

e) The product of A x B is:

a) to solve the first item, we need to look in the table and find the corresponding numbers.

The characters are:

H E L L O _ R A Y A

The corresponding string is 8, 5 12, 12, 15, 0, 18, 1, 25, 1

b) This string will be converted in 5 2x1 matrices, because there are 10 elements in total, and the matrices have 2 elements each:

`(10)/(2)=5`

c) Now, to write the 2x1 matrices, we need to take the string of numbers in pairs and create the 5 matrices:

`\begin{bmatrix}{8} & {} \\ {5} & {}\end{bmatrix}\begin{bmatrix}{12} & {} \\ {12} & {}\end{bmatrix}\begin{bmatrix}{15} & {} \\ {0} & {}\end{bmatrix}\begin{bmatrix}{18} & {} \\ {1} & {}\end{bmatrix}\begin{bmatrix}{25} & {} \\ {1} & {}\end{bmatrix}`

d) If we join all the 2x1 matrices, we'll get a 2x5 matrix. This is matrix B.

`\begin{gathered} 8\text{ 1}2\text{ 1}5\text{ 1}8\text{ 2}5 \\ 5\text{ }12\text{ }0\text{ }1\text{ }1 \end{gathered}`

e) The product of matrices is: