Question

Matrix multiplication was used to encode a message using the given encoding matrix: [i 47 A=1 |-1 -3] The original message was converted to row matrices of size: 1x 2 and each was multiplied by A. sp 0 A 1 B 2 C 3 D 4 E F G H I J K L M N O P Q R 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 14 | 15 | 16 | 17 | 18 S T U V W X Y Z 19 20 21 22 23 24 25 26 Coded message: 11 52 -8 -9 -13 -39 5 20 12 56 5 20 -2 7 9 41 25 100 Find A and use it to decode the message

Answer 1

Answer:

$A^{-1}=\left[\begin{array}{cc}-3&-4\\1&1\end{array}\right]$ message SHOW_ME_THE_MONEY_  

Step-by-step explanation:

The matrix

$A=\left[\begin{array}{cc}1&4\\-1&-3\end{array}\right]\rightarrow |A|=(1 \times -3)-(-1\times 4)=1\\\rightarrow A^{-1}=\left[\begin{array}{cc}-3&-4\\1&1\end{array}\right]$

We can check that in fact A*A^⁻1=I_2 the identity matrix of size 2 x 2.

Now the message was divided in 1 x 2 matrices, then we have that the sequence given is the result of multiplying m by A, so to get m again we multiply now by A^⁻1. and we get the next table

Encoded message    Decoded message    message in letters by association

11 52        19 8    S H

-8 -9         15 23  O W

-13 -39        0 13   _ M

5 20         5 0   E _

12 56        20 8    T H

5 20         5 0    E _

-2 7           13 15  M O

9 41         14 5   N E

25 100       25 0    Y _

Then the message decoded is SHOW_ME_THE_MONEY_