Application of Linearity
A. Translating...
In order to translate the matrix sent to him by Alice, Bob should separate the encoded matrix into sets of 3 characters and then multiply it by the inverse of the matrix originally provided to him by Alice.
B. Eve-sdropping
Without the inverse of the matrix, there isn't any way that Eve could figure out what Bob and Alice were talking about. However, she has the necessary information to figure out what the inverse of the matrix is.
C. Decrypting
First you have to find the inverse of the matrix. It turns out to be:
$ A=\left[ \begin{array}{ccc} -\frac{2}{3} & 0 & \frac{2}{3} \\ 0 & 1 & 0 \\ 4 & 0 & -1 \end{array} \right] \rightarrow A^{-1}= \left[ \begin{array}{ccc} \ \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 & 0 & \frac{1}{3} \end{array} \right] $
After you find the inverse of the matrix, multiply the inverse by [ 2 23 3 ] to find the encoded message.
After decoding the matrix, I got the letters "BWE"
