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This correspondence describes a method for finding a closed-form expression for the inverse of some Gram matrices that occur in least-square theory. The method is applicable if an analytic expression for an appropriate set of orthonormal functions can be found. In particular, the method provides a new derivation for the inverse of the generalized Hilbert matrix. An explicit formula is also given for a class of polynomials that is orthonormal over any finite interval with respect to the uniform weight function.