  In this paper, we first present an explicit expression for the inverse\emph{}
of a type of matrices. As special applications, the inverse of some matrices
arising from implicit time integration techniques, such as the well-known
implicit Runge-Kutta schemes and block implicit methods, can also be explicitly
determined. Adiitionally, we introduce three fast algorithms for computing the
elements of the inverse of these matrices in $O(n^2)$ arithmetic operations,
i.e., the first one is based on Traub algorithm for fast inversion of
Vandermonde matrices, while the other two utilize the special structure of the
matrices. Finally, some symbolic and numerical results are presented to show
that our algorithms are both highly efficient and accurate.
