How can Bob decrypt Alice's message?
Bob can decrypt the message Alice sent him by inverting the matrix he received, partitioning the message into vectors of 3x1 and multiplying the vectors by the inverted matrix.
Can Eve decrypt the message without finding the inverse of the secret matrix?
Yes, with the input and output of the system, Eve can solve for the matrix used via a simple set of simultaneous equations.
$ \begin{bmatrix} 1 & 0 & 4 \end{bmatrix} \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} $ = $ \begin{bmatrix} 2 & 0 & 0 \end{bmatrix} $
The above is an example of one vector equation. Continuing this example to all three vectors will give you a set of equations that can be solved for the unknown matrix.
The unknown matrix is:
$ \begin{bmatrix} -2/3 & 0 & 4 \\ 0 & 1 & 0 \\ 2/3 & 0 & -1 \end{bmatrix} $
