How can Bob decrypt the message?
Bob can easily decrypt the message. The first thing he needs to do is to find the inverse of the secret matrix that Alice tells him. Then he can simply multiply the inverse matrix that he finds to the encrypted message, and change the numbers to the corresponding letters to obtain the message.
Can Eve decrypt the message without finding the inverse of the secret matrix?
I don't think it's possible for Eve to obtain the message without finding the inverse of the secret matrix. She needs to know what the secret matrix is in order to decrypt the message.
What is the decrypted message corresponding to (2,23,3)?
Using the information given above, we can create two matrix, one for (1,0,4,0,1,0,1,0,1) and one for (2,0,0,0,1,0,0,0,3). Using matlab with the following code, we multiply the second matrix with the inverse of the first one to yield the third matrix. Finally we take the inverse of the third matrix and multiply to the given message. Translating the message using the order of alphabets yields "BWE"
x = [2 0 0; 0 1 0; 0 0 3] y = [1 0 1; 0 1 0; 4 0 1] invy = inv(y) z = x*invy invz = inv(z) message = [2; 23; 3] decrypted = invz*message %output is [2; 23; 5]
