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We construct a neutrino mass matrix $M_{\nu}$ via a seesaw mechanism whithperturbed invariant under a cyclic permutation by introducing one parameter$\delta$ into the diagonal elements of $M_{\nu}$ with assumption that trace ofthe perturbed $M_{\nu}$ is equal to trace of the unperturbed $M_{\nu}$. Wefound that the perturbed neutrino mass matrices $M_{\nu}$ can predicts themass-squared difference $\Delta m_{ij}^{2}\neq 0$ with the possible hierarchyof neutrino mass is normal or inverted hierarchy. By using the advantages ofthe mass-squared differences and mixing parameters data from neutrinooscillaton experiments, we obtained neutrino masses in inverted hierarchy withmasses: $|m_{1}|=0.101023$ eV, $|m_{2}|=0.101428$ eV, and $|m_{3}|=0.084413$eV.

Perturbed invariant under a cyclic permutation with trace of neutrino mass matrix remain constant