How Can Bob Decrypt the Message
Bob can decrypt the message quite easily. Since Alice sends him the secret matrix she uses to encrypt the message, Bob can take that matrix and use its inverse to decrypt the message. All he needs to do is take the inverse of the secret matrix, multiply it to the message vector three elements at a time, then match each number with its associating letter to get the original message.
Can Eve Decrypt the Message Without Finding the Secret Matrix
No, she will not be able to decrypt the message without finding the secret matrix. However, since she has the message and its decrypted output, she can set up a system of equations to easily find the secret matrix.
What is the decrypted message
Using MATLAB, I found the secret matrix to be: $ \,\begin{matrix} - \frac{2}{3} & 0 & \frac{2}{3} \\ 0 & 1 & 0 \\ 4 &0 & -1 \end{matrix} $
Then, takings its inverse gives you:
$ \begin{matrix} \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 &0 & \frac{1}{3} \end{matrix} $
Then multiply the encrypted message matrix by the inverse of the secret matrix and you get: $ \begin{matrix} 2 & 23 & 5\end{matrix} $
which corresponds to B W E if you match the numbers with their corresponding letters.
