   # Encryption 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 &lt;space&gt; = 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 [ 1 2 3 4 ] . 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 = [ 1 2 3 4 ] [ 1 15 20 13 19 9 14 2 18 0 9 19 15 0 ] = [ 5 51 20 31 57 39 14 11 117 60 75 133 87 42 ] Which we can also write as [ 5 11 51 117 20 60 31 75 57 133 39 87 14 42 ] To decipher the encoded message, multiply the encrypted matrix by A − 1 . Exercises 63–66 use the above matrix A for encoding and decoding. Decode the following message, which was encrypted using the matrix A . [ 33 69 54 126 11 27 20 60 29 59 65 149 41 87 ] ### Finite Mathematics

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337280426

#### Solutions

Chapter
Section ### Finite Mathematics

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337280426
Chapter 4.3, Problem 65E
Textbook Problem
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## Encryption 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 [ 1 2 3 4 ] . 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 = [ 1 2 3 4 ] [ 1 15 20 13 19 9 14 2 18 0 9 19 15 0 ] = [ 5 51 20 31 57 39 14 11 117 60 75 133 87 42 ] Which we can also write as [ 5 11 51 117 20 60 31 75 57 133 39 87 14 42 ] To decipher the encoded message, multiply the encrypted matrix by A − 1 . Exercises 63–66 use the above matrix A for encoding and decoding.Decode the following message, which was encrypted using the matrix A. [ 33 69 54 126 11 27 20 60 29 59 65 149 41 87 ]

To determine

To calculate: The decoded form of the message  where each letter in the alphabet is represented by a number, and the sequence of numbers is arranged in the form of a matrix with two rows and the sequence is multiplied on the left by A where A=.

### Explanation of Solution

Given Information:

Each letter in the alphabet is represented by a number, and the sequence of numbers is arranged in the form of a matrix with two rows and the sequence is multiplied on the left by A where A=.

Formula used:

A matrix with m rows and n columns is of dimension m×n, where m and n are positive integers.

Calculation:

Consider the encoded message,



Write as a matrix with two rows as below,



To calculate the decoded message, multiply the inverse of the matrix A by the encoded message.

Thus, decoded message is 1.

Since, number of rows in 1 is 2 and number of columns is 2. Thus, dimension of 1 is 2×2.

Since, number of rows in  is 2 and number of columns is 7. Thus, dimension of  is 2×7.

Since, number of columns in 1 is equal to the number of rows in , the product 1 is defined

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