Does the system matrix being singular tell us anything about the system?
If the system matrix $A$ is singular then it has at least one eigenvalue equal to $0$. Since the eigenvalues of $A$ represent the poles of the transfer function of the system, there is at least one pole at $s=0$, which makes the system unstable (or marginally stable if there is a single pole at $s=0$, but in any case not asymptotically stable).