  The results on Vandermonde-like matrices were introduced as a generalization
of polynomial Vandermonde matrices, and the displacement structure of these
matrices was used to derive an inversion formula. In this paper we first
present a fast Gaussian elimination algorithm for the polynomial
Vandermonde-like matrices. Later we use the said algorithm to derive fast
inversion algorithms for quasiseparable, semiseparable and well-free
Vandermonde-like matrices having $\mathcal{O}(n^2)$ complexity. To do so we
identify structures of displacement operators in terms of generators and the
recurrence relations(2-term and 3-term) between the columns of the basis
transformation matrices for quasiseparable, semiseparable and well-free
polynomials. Finally we present an $\mathcal{O}(n^2)$ algorithm to compute the
inversion of quasiseparable Vandermonde-like matrices.
