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The generic predictions of inverse Majorana neutrino mass matrix,$M_{\nu}^{-1}$ with texture two zeros, in the basis where charged lepton massmatrix is diagonal, have been obtained for neutrino masses and mixings,especially, $\theta_{13}$ and CP-violating phases. Such type of mass models arenatural in the context of seesaw models with Dirac neutrino mass matrix,$M_{D}$, diagonal. Out of the fifteen possible texture two zeros patterns ofthe inverse neutrino mass matrix only seven are found to be compatible with theavailable data on neutrino masses and mixings including the latest $T\emph{2}K$observation of non-zero $\theta_{13}$. It is, also, found that $\theta_{13}=0$is disallowed in the mass models investigated in the present work. While theneutrino mass matrices of type I and II are found to be necessarily CPviolating, type III is found to be CP-violating for a special case where$\theta_{23}$ lie above maximality. The atmospheric mixing angle, $\theta_{23}$is found to dictate the possible hierarchies in type I and II neutrino massmodels. For type III, additional information regarding $m_{ee}$ will berequired to rule out the inverted hierarchy (IH). A maximal $\theta_{23}$ isfound to be disallowed in all types of texture two zero $M_{\nu}^{-1}$Ans$\ddot{a}$tz.

Non-zero $\theta_{13}$ and CP-violation in inverse neutrino mass matrix