How Can Bob Decrypt the Message?
Bob can decrypt the message by multiplying the encrypted vector by the inverse of the matrix used to encrypt the message. Then by replacing the numbers with their corresponding letters, and the 0s with spaces he can read the intended message.
Can Eve Decrypt the Message without Finding the Inverse Matrix?
The answer is no, she cannot decrypt it without finding the inverse matrix. However, she can use the input and output vectors to find the encryption matrix and then find its inverse to decrypt the encrypted message (output vector). (Although in her situation she doesn't need to find the encryption matrix or decrypt the message at all, she already knows the un-coded message (the input-in vector form). So all she needs to know is the algorithm that replaces the letters with numbers in order to intercept the message.)
Decrypting the Message
The secret encryption matrix was found to be: $ \begin{bmatrix} -0.6667 & 0 & 0.6667 \\ 0 & 1.0000 & 0 \\ 4.0000 & 0 & -1.0000 \\ \end{bmatrix} $
Its inverse is given by:
$ A^{-1} = \begin{bmatrix} 1/2 & 0 & 1/3 \\ 0 & 1 & 0 \\ 2 & 0 & 1/3 \end{bmatrix} $
When given the encrypted vector <2,23,3> , multiplying it by $ A^{-1} $ one obtains the decrypted vector <2,23,5>. Then after replacing each number by its corresponding letter in the alphabet we get find that the message is
"BWE"
