The matrices are correct; but not the matrix that is given on the cited angelfire.com page even though the way of derivation that is given there is correct. there only the ab prefactor of one term of c3 has to be removed. 
possible citation for the correct matrix: TABMathematical techniques in crystallography and material science / Edward Prince / Springer 1994 (see chapter "metric tensor").
the definitions of the cited angelfire page are equal to the definitions here, there is just one error in calculus.

by the way: no chemist is needed here. every scientist can proof these matrices to be correct.

129.69.120.176