How to decrypt the message
Since bob knows that the message is split into groups of 3, goes into a 3 by 3 matrix, and being outputted as the encrypted message, he can just decrypt the message by splitting his encrypted message into groups of 3 and multiply it with the inverse of the 3 by 3 matrix
Can the message be decrypted without using inverse of matrix?
Possibly. Besides the inverse of matrix, i believe by using eiginvectors and eiginvalues, the code can be decrypted. However, i'm not sure how can it be done.
What is the decrypted message
As $ \left[ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right] \times \left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \end{array} \right] = \left[ \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array} \right] $, The secret matrix $ A = \left[ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right] = \left[ \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array} \right] \times \left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \end{array} \right]^{-1} = \left[ \begin{array}{ccc} -\frac{2}{3} & 0 & \frac{2}{3} \\ 0 & 1 & 0 \\ 4 & 0 & -1 \end{array} \right] $
Therefore the inverse of matrix $ A^{-1} = \left[ \begin{array}{ccc} -\frac{2}{3} & 0 & \frac{2}{3} \\ 0 & 1 & 0 \\ 4 & 0 & -1 \end{array} \right]^{-1} = \left[ \begin{array}{ccc} \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 & 0 & \frac{1}{3} \end{array} \right] $
Thus the decrypted message is $ A^{-1} \times \left[ \begin{array}{c} 2\\ 23\\ 3\end{array} \right] = \left[ \begin{array}{ccc} \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 1 & 0 \\ 2 & 0 & \frac{1}{3} \end{array} \right] \times \left[ \begin{array}{c} 2\\ 23\\ 3 \end{array} \right] = \left[ \begin{array}{c} 2\\ 23\\ 5 \end{array} \right] $
The decrypted message is B, W, E
