  This paper discusses the explicit inverse of a class of seven-diagonal (near)
Toeplitz matrices, which arises in the numerical solutions of nonlinear
fourth-order differential equation with a finite difference method. A
non-recurrence explicit inverse formula is derived using the Sherman-Morrison
formula. Related to the fixed-point iteration used to solve the differential
equation, we show the positivity of the inverse matrix and construct an upper
bound for the norms of the inverse matrix, which can be used to predict the
convergence of the method.
