Neutrino Mass Matrices with Vanishing Determinant and $\theta_{13}$

We investigate the prospects for scenarios with vanishing determinant neutrino mass matrices and vanishing $\theta_{13}$ mixing angle. Normal and inverse mass hierarchies are considered separately. For normal hierarchy it is found that neutrinoless double beta decay cannot be observed by any of the present or next generation experiments. For inverse hierarchy the neutrinoless double beta decay is, on the contrary, accessible to experiments. We also analyse for both hierarchies the case for texture zeros and equalities between mass matrix elements. No texture zeros are found to be possible nor any such equalities, apart from the obvious ones.