  A novel factorization for the sum of two single-pair matrices is established
as product of lower-triangular, tridiagonal, and upper-triangular matrices,
leading to semi-closed-form formulas for tridiagonal matrix inversion.
Subsequent factorizations are established, leading to semi-closed-form formulas
for the inverse sum of two single-pair matrices. An application to derive the
symbolic inverse of a particular Gram matrix is presented, and the numerical
stability of the formulas is studied.
