Let's consider the given cryptogram!
21 - 11 14 29 - 11 - 18 32 - 6 - 26 31 - 19 - 12 10 6 26 13 - 11 -2 37 28 - 8 5 13 36
We want to decode the cryptogram using the decoding matrix A^(- 1). We can do so in three steps.
Find the decoding matrix which is the inverse of the given matrix A.
Partition the message into groups of three to form the coded row matrices.
Multiply each coded row matrix by A^(- 1).
We can do each of these steps one at a time.
Finding the Decoding Matrix
We can find the decoding matrix using a graphing calculator. First, we can look at the given matrix A.
A= 1&- 1&0 0&1&2 1&1&- 2
Now, let's enter the matrix A into the calculator. Push 2ND and x^(- 1). Then, using the right arrow, navigate to the Edit tab and push ENTER. Define the dimensions and enter the elements.
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Now that we have entered the matrix we push 2ND followed by MODE to go back to the main screen. Then, we push 2ND and x^(- 1), select our matrix, and push ENTER.
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Finally, we push x^(- 1), followed by ENTER. This will return us the inverse of our matrix!
We cannot properly see the elements of the inverse matrix because they are written as decimals. Let's express them as fractions. To do so, press MODE, scroll down to the Answers option, and select Frac. Then, we go back to the main screen by pushing 2ND and MODE, and push ENTER one last time.
We found the decoding matrix!
A^(-1) =
23 & 13 & 13 [0.4em]
- 13 & 13 & 13 [0.4em]
16 & 13 & - 16 [0.4em]
Partition the Message
Now, we can partition the message into groups of three.
ccc 21& - 11& 14 29& - 11 & - 18 32& - 6& - 26 31& - 19 & - 12 10& 6& 26 13& - 11 & -2 37& 28 & - 8 5 & 13 & 36
Let's rewrite each row as a 1* 3 matrix.
1{21&- 11&14 29& - 11 & - 18
We want to decode our cryptogram.
21&- 11&14 29& - 11 & - 18 32 &- 6&- 26 31& - 19&- 12 10& 6&26 13& - 11&- 2 37& 28&- 8 5& 13&36
We can do so by multiplying each coded row matrix by A^(- 1). Let's start with the first coded row matrix!
Now, we can list the sequence of coded row matrices.
9&0&23 9&12&12 0&2&5 0&2&1 3&11&0
Let's recall which numbers are assigned to which letter in the alphabet. Remember that blank space corresponds to 0.
Code
0=
9=I
18=R
1=A
10=J
19=S
2=B
11=K
20=T
3=C
12=L
21=U
4=D
13=M
22=V
5=E
14=N
23=W
6=F
15=O
24=X
7=G
16=P
25=Y
8=H
16=Q
26=Z
Finally, we can use our table to rewrite each 1* 3 matrix.
Group of Three Letters
Coded Row Matrix
20&8&1
T&H&A
20&0&9
T& &I
19&0&13
S& &M
25&0&6
Y & & F
9&14&1
I&N&A
12&0&1
L & & A
14&19&23
N&S&W
5&18&0
E&R&
Now, we can list the sequence of decoded row matrices.
T&H&A T& &I S& M Y& &F I&N&A L& &A N&S&W E&R&
Finally, we can remove the matrix notation and write the message corresponding to our cryptogram.
THAT IS MY FINAL ANSWER
