Matrix very difficult - Hard level.

[henriqueheck]

An own code given by the resolution of the matrix product AB = C, where B is a square matrix of Order 2, and each letter of the alphabet corresponds to a number, that is: a = 1, b = 2, c = 3 ... z = 26, according to
Table below.
b=
3 1
2 1

Unmute the message:
70 25 102 39 75 28 49 21 53 19 41 20 51 23 58 21 82 34 25 10 30 13 11 04 63 24

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2

 

[HallsofIvy]

It is not clear to me exactly what this code is. You say "An own code given by the resolution of the matrix product AB = C" Since B is a 2 by 2 matrix A would have to be a 2 by 1 matrix. Are you "coding" two letters at a time? For example, if the first word in the message were "The" would you represent the first two letters as \(\displaystyle \begin{bmatrix} 20 & 8\end{bmatrix}\)? And so those two letters would be coded as \(\displaystyle \begin{bmatrix}20 & 8 \end{bmatrix}\begin{bmatrix}3 & 1 \\ 2 & 1 \end{bmatrix}= \begin{bmatrix} 76& 28\end{bmatrix}\)?

If that is your coding scheme then break the given code into groups of two numbers and "uncode" them two numbers at a time. Since C= AB, \(\displaystyle A= CB^{-1}\).

Here, \(\displaystyle B= \begin{bmatrix}3 & 1 \\ 2 & 1 \end{bmatrix}\) has determinant 1 so it is easy to see that \(\displaystyle B^{-1}= \begin{bmatrix}1 & -1 \\ -2 & 3\end{bmatrix}\). The first two numbers in the code are "70 25", so if my interpretation is correct, they would decode as \(\displaystyle \begin{bmatrix}70 & 25 \end{bmatrix}\begin{bmatrix}1 & -1 \\ -2 & 3 \end{bmatrix}= \begin{bmatrix}20 & 5 \end{bmatrix}\) or "TE".