Raj encoded a secret phrase using matrix multiplication. Using $A=1, B=2, C=3$, and so on, he multiplied the clear text code for each letter by the matrix $C=\left[\begin{array}{ll}2 & 5 \\ 1 & 2\end{array}\right]$ to get a matrix that represents the encoded text. The matrix representing the encoded text is $\left[\begin{array}{ccccccc}85 & 111 & 135 & 111 & 95 & 101 & 153 \\ 38 & 46 & 55 & 45 & 41 & 44 & 64\end{array}\right]$. What is the secret phrase? Determine the location of spaces after you decode the text.
A. YUMMY IS THE CORN
B. THE TOMATO IS RED
C. THE CORN IS YUMMY
D. RED IS THE TOMATO
Answer (2)
Find the inverse of the encoding matrix C .
Multiply the encoded matrix by the inverse of C to get the decoded matrix.
Convert the numbers in the decoded matrix to letters.
Insert spaces to form the secret phrase: T H ECORN I S Y U MM Y .
Explanation
Understanding the Problem We are given an encoded matrix and an encoding matrix C . Our goal is to decode the matrix and find the secret phrase. The encoding is done by multiplying the clear text code by the matrix C . Therefore, to decode, we need to multiply the encoded matrix by the inverse of C .
Finding the Inverse of the Encoding Matrix First, we need to find the inverse of the encoding matrix C = [ 2 1 5 2 ] . The inverse of a 2x2 matrix [ a c b d ] is given by a d − b c 1 [ d − c − b a ] . In our case, a = 2 , b = 5 , c = 1 , d = 2 . So, a d − b c = ( 2 ) ( 2 ) − ( 5 ) ( 1 ) = 4 − 5 = − 1 . Therefore, the inverse of C is C − 1 = − 1 1 [ 2 − 1 − 5 2 ] = [ − 2 1 5 − 2 ] .
Decoding the Matrix Next, we multiply the encoded matrix [ 85 38 111 46 135 55 111 45 95 41 101 44 153 64 ] by the inverse of C , which is [ − 2 1 5 − 2 ] .
The resulting matrix is:
[ − 2 1 5 − 2 ] [ 85 38 111 46 135 55 111 45 95 41 101 44 153 64 ] = [ ( − 2 ) ( 85 ) + ( 5 ) ( 38 ) ( 1 ) ( 85 ) + ( − 2 ) ( 38 ) ( − 2 ) ( 111 ) + ( 5 ) ( 46 ) ( 1 ) ( 111 ) + ( − 2 ) ( 46 ) ( − 2 ) ( 135 ) + ( 5 ) ( 55 ) ( 1 ) ( 135 ) + ( − 2 ) ( 55 ) ( − 2 ) ( 111 ) + ( 5 ) ( 45 ) ( 1 ) ( 111 ) + ( − 2 ) ( 45 ) ( − 2 ) ( 95 ) + ( 5 ) ( 41 ) ( 1 ) ( 95 ) + ( − 2 ) ( 41 ) ( − 2 ) ( 101 ) + ( 5 ) ( 44 ) ( 1 ) ( 101 ) + ( − 2 ) ( 44 ) ( − 2 ) ( 153 ) + ( 5 ) ( 64 ) ( 1 ) ( 153 ) + ( − 2 ) ( 64 ) ] = [ 20 9 8 19 5 25 3 21 15 13 18 13 14 25 ] .
Converting Numbers to Letters Now, we convert the numbers in the decoded matrix to letters using A=1, B=2, ..., Z=26. The decoded matrix is [ 20 9 8 19 5 25 3 21 15 13 18 13 14 25 ] .
This corresponds to the letters:
[ T I H S E Y C U O M R M N Y ] .
Combining these letters, we get the phrase "THECORNISYUMMY".
Inserting Spaces Finally, we need to insert spaces in the decoded text to form a meaningful phrase. From the given options, the correct phrase is "THE CORN IS YUMMY".
Final Answer Therefore, the secret phrase is "THE CORN IS YUMMY".
Examples
Matrix multiplication can be used in cryptography to encode and decode secret messages. By using a specific matrix as a key, the original message can be transformed into an unreadable format. Only someone with the inverse of the key matrix can decode the message back to its original form. This method is used to secure communications in various applications, such as online banking, email encryption, and secure file transfer.
The secret phrase is decoded by finding the inverse of the encoding matrix, multiplying it by the encoded text, and converting the resulting numbers into letters. The final decoded phrase is "THE CORN IS YUMMY", which corresponds to option C. Therefore, the answer is C. THE CORN IS YUMMY.
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