1. How can Bob decrypt the message?
According the problem, there is a secret matrix, X, that is known to Bob. Bob has also been sent the encrypted matrix as well. If you set it up as an equation as follows:
$ \left[ \begin{array}{ccc} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array} \right] \times \left[ \begin{array}{ccc} X \end{array} \right] = \left[ \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array} \right] $
Then, Bob can simply just find the inverse of the secret matrix (X) and multiply it with the encrypted matrix.
2. Can Eve decrypt the message without finding the inverse of the secret matrix?
I would say no, but it should be possible if she knows the properties of linearity well. I guess it depends on if she is an electrical engineer.
3. What is the decrypted message corresponding to (2,23,3)? (Write it as a text.)
Using MATLAB to do some matrix math, I found the secret matrix to be: $ [X] = \left[ \begin{array}{ccc} \ \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 & 0 & \frac{1}{3} \end{array} \right] $.
Now, just multiply the secret matrix with the column vector [2,23,3]:
$ [X]^{-1} \times \left[ \begin{array}{ccc} 2\\ 23\\ 3\end{array} \right] = \left[ \begin{array}{ccc} 2\\ 23\\ 5\end{array} \right] $
Which translates to the letters BWE.
