How to decode such messages?

Question

In the Matrix encryption method we transform message into matrix and encrypt it by pre-multiplying it with some encoding matrix.
The decoding is based on the fact that the inverse of decoding matrix can bring back the original message as they cancel out.
But what if the encoding matrix is singular ie. Its Inverse does not exist.
Then how will we find the original message Would it be possible to decode the message.

Answer 1

In general, if an encryption method is represented by a function $f$ mapping source messages to encrypted messages, it's decodable if and only if $f$ is injective. Otherwise, if both $f(x_1)=y$ and $f(x_2)=y$, it's not possible to decode the message $y$ even in principle, since either $x_1$ or $x_2$ could have generated it.

In your case, if $A$ is a singular matrix, then it cannot be injective, so there is no way of decoding the corresponding message.