Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take < space > = 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes[1  2  15  18  20  0  13  9  19  19  9  15  14].To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 2 × 2 matrix  34 21. We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A.Encrypted Matrix =  34 21 115201319914 2180919150  =  1111760751338742 4484035573328,which we can also write as[11  4  117  48  60  40  75  35  133  57  87  33  42  28].To decipher the encoded message, multiply the encrypted matrix by A−1. The following question uses the above matrix A for encoding and decoding.Decode the following message, which was encrypted using the matrix A. (Include any appropriate spaces in your answer.)[69  21  126  54  27  13  60  40  59  16  149  61  87  28]

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Asked Nov 12, 2019
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Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take < space > = 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes

[1  2  15  18  20  0  13  9  19  19  9  15  14].

To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 2 × 2 matrix 

 
3 4
 
2 1
.

 We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A.

Encrypted Matrix  = 
 
3 4
 
2 1
 
1 15 20 13 19 9 14
 
2 18 0 9 19 15 0
   = 
 
11 117 60 75 133 87 42
 
4 48 40 35 57 33 28
,

which we can also write as

[11  4  117  48  60  40  75  35  133  57  87  33  42  28].

To decipher the encoded message, multiply the encrypted matrix by 

A−1.

 The following question uses the above matrix A for encoding and decoding.

Decode the following message, which was encrypted using the matrix A. (Include any appropriate spaces in your answer.)

[69  21  126  54  27  13  60  40  59  16  149  61  87  28]
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Expert Answer

Step 1

Consider the given question,

Find the inverse of the matrix A,

A =
2
1
adj4
Since, A
(cofactorA)
|4
1
-4
So, A
3 4
2 1
1
-4
-2
А
-5
4
5
5
thus, A
2
5
5
help_outline

Image Transcriptionclose

A = 2 1 adj4 Since, A (cofactorA) |4 1 -4 So, A 3 4 2 1 1 -4 -2 А -5 4 5 5 thus, A 2 5 5

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Step 2

Multiply the Encrypted matrix wit...

Encrypted matrix is
[69 21
28]
126 54 27 13
60 40 59 16 149 61 87
it is also written as
69 126 27 60 59 149 87
28
21
54
13 40 16
61
69 126 27 60
59 149 87
Now, A
21
54
13 40
16
61
28
1
69 126 27 60 59 149 87
5
2
3 21
54
13 40 16
61
28
5
5
18 5 20
1
19
5
0 14 23 18
15 18 3
and, it is written as
5 18]
3 15 18 18 5 3 20 0 1 14
19 23 5
|15
help_outline

Image Transcriptionclose

Encrypted matrix is [69 21 28] 126 54 27 13 60 40 59 16 149 61 87 it is also written as 69 126 27 60 59 149 87 28 21 54 13 40 16 61 69 126 27 60 59 149 87 Now, A 21 54 13 40 16 61 28 1 69 126 27 60 59 149 87 5 2 3 21 54 13 40 16 61 28 5 5 18 5 20 1 19 5 0 14 23 18 15 18 3 and, it is written as 5 18] 3 15 18 18 5 3 20 0 1 14 19 23 5 |15

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