According to the differentiation method of the trace of matrices and the symmetric property of the squared matrix, L = B TPC, extremum approaches in some different fields were examined as examples. The examples examined were: Jug's method on bond order; Murrell's two theorems; Golebiewski's statement on the maximum overlap method of hybridization; Lykos and Schmeising's maximum overlap orbital; the derivation of the Roothaan's equation in the closed-shell or open-shell approach; and the natural expansion of the wavefunction and of the qth order reduced density matrix. The results indicate that the trace algebra, from which the differentiation method is constructed, will be an important and powerful tool in the future development of quantum chemistry.