How can Bob decrypt the message?
All Bob has to do to decrypt the message is make a matrix out of encrypted vector and multiply by the inverse of the secret matrix. The result will be the decoded vector and all Bob then has to do is substitute letters for the numbers.
Can Eve decrypt the message without finding the inverse of the secret matrix?
No, but with the data she has she can easily find the inverse of the secret matrix.
What is the decrypted message corresponding to (2,23,3)?
First, we get the secret matrix.
$ \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix} * \begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}^{-1} = \begin{bmatrix} \frac{-2}{3} & 0 & \frac{2}{3} \\ 0 & 1 & 0 \\ 4 & 0 & -1 \end{bmatrix} $
Now, we take the inverse of the secret matrix.
$ \begin{bmatrix} \frac{-2}{3} & 0 & \frac{2}{3} \\ 0 & 1 & 0 \\ 4 & 0 & -1 \end{bmatrix}^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 & 0 & \frac{1}{3} \end{bmatrix} $
Now with the inverse we just multiply by the encoded vector and get our decoded vector which we translate to text.
$ \begin{bmatrix} \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 & 0 & \frac{1}{3} \end{bmatrix} $ * $ \begin{bmatrix} 2 \\ 23\\ 3 \end{bmatrix} = \begin{bmatrix} 2 \\ 23\\ 5 \end{bmatrix} $
In text, the decoded vector is BWE.
