How can Bob decrypt the message?

Since Alice gives the encryptor matrix, to make it a decryptor matrix Bob will need to invert the matrix. Then, multiply it with the code so it will be decrypted. After the numbers come out, he will need to change each number with the respective alphabet character.

Can Eve decrypt the message without finding the inverse of the secret matrix?

Without finding the inverse of the secret matrix there is no way for Eve to know the message except she is a magician. But if she wanted to find one, she can see this equation:

$ \begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} A & B & C \\ D & E & F \\ G & H & I \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix} $

Solving for the alphabet characters matrix, she will find:

$ \begin{bmatrix} -\frac{2}{3} & 0 & \frac{2}{3} \\ 0 & 1 & 0 \\ 4 & 0 & -1 \end{bmatrix} $