  This paper is a continuation of the work on unbounded Toeplitz-like operators
$T_\Om$ with rational matrix symbol $\Om$ initiated in Groenewald et. al
(Complex Anal. Oper. Theory 15, 1(2021)), where a Wiener-Hopf type
factorization of $\Om$ is obtained and used to determine when $T_\Om$ is
Fredholm and compute the Fredholm index in case $T_\Om$ is Fredholm. Due to the
high level of non-uniqueness and complicated form of the Wiener-Hopf type
factorization, it does not appear useful in determining when $T_\Om$ is
invertible. In the present paper we use state space methods to characterize
invertibility of $T_\Om$ in terms of the existence of a stabilizing solution of
an associated nonsymmetric discrete algebraic Riccati equation, which in turn
leads to a pseudo-canonical factorization of $\Om$ and concrete formulas of
$T_\Om^{-1}$.
