Can Bob decrypt the message?
Yes Bob can decrypt the message by finding the inverse of the encryption matrix(which he has the information to do) and multiplying the given matrix by it.
Can Eve decrypt the message without finding the inverse of the secret matrix?
No Eve cannot decrypt the message without finding the inverse of the secret matrix. She does however have enough information to find the inverse of the matrix.
What is the decrypted message corresponding to(2,23,3)?
We can find Inverse A:
$ 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] $
Once we have Inverse A, we know that:
$ A^{-1}*M=DM $
Where M is the message (2,23,3) and DM is the decrypted message.
$ \left[ \begin{array}{ccc} \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 & 0 & \frac{1}{3} \end{array} \right] \left[ \begin{array}{c} 2\\ 23\\ 3\end{array} \right] = \left[ \begin{array}{c} 2\\ 23\\ 5 \end{array} \right] $
The decrypted message is (2,23,5) which is BWE.
