The cryptogram decoding using encoded 2×2 matrix.

Answer to Problem 59E
The final message is MEET ME TONIGHT RON
Explanation of Solution
Given information:
8 21 −15 −10 −13 −13 5 10 5 255 19 −1 6 20 40 −18 −18 1 16
Formula used:
Matrix multiplication is used.
Calculation:
To decode the following cryptogram:
8 21 -15 -10 -13 -13 5 10 5 25 5 19 -1 6 20 40 -18 -18 1 16
Using the some unknown decoding matrix A−1=[abcd]
Dividing them into 1×2 matrices we get,
[821][−15−10][−13−13][510][525][519][−16][2040][−18−18][116]
The last word of the message is _RON.
So,
[−18−18]×A−1=[−1818]×[abcd] =[−18a−18c−18b−18d]
The first two letters of the last word are _ & R. So,
−18a−18c=0 a=−c
And
−18b−18d=18 b+d=−1
Also ,
[116]×A−1=[116]×[abcd] =[a+16cb+16d]
The last two letters of the last word are 0 & N. So,
a+16c=15 & b+16d=14
Using a=−c and b+d=−1 we get
−c+16c=15 15c=15 c=1
a=−1
Thus, get
−1−d+16d=14 15d=15 d=1
b=−2
Thus, get
A−1=[−1−211]
Now to get the uncoded row matrices,
Coded matrix Decoding matrix Uncoded Matrix
[821][−1−211][135][−15−10][−1−211][520][−13−13][−1−211][013][510][−1−211][50]
[525][−1−211][2015][519][−1−211][149][−16][−1−211][78][2040][−1−211][200]
We get the following uncoded row matrices,
[135][520][013][50][2015][149][78][200]
Splitting them we get,
13 5 5 20 0 13 5 0 20 15 14 9 7 8 20 0
When the numbers are converted to the alphabets they are assigned to we get the message
MEET ME TONIGHT
Using the last word of the message, the final message is
MEET ME TONIGHT RON
Conclusion:
The final message is MEET ME TONIGHT RON
Chapter 8 Solutions
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