  We investigate the prospects for neutrinoless double beta decay, texture
zeros and equalities between neutrino mass matrix elements in scenarios with
vanishing determinant mass matrices for vanishing and finite $\theta_{13}$
mixing angle in normal and inverse mass hierarchies. For normal hierarchy and
both zero and finite $\theta_{13}$ it is found that neutrinoless double beta
decay cannot be observed by any of the present or next generation experiments,
while for inverse hierarchy it is, on the contrary, accessible to experiments.
Regarding texture zeros and equalities between mass matrix elements, we find
that in both normal and inverse hierarchies with $\theta_{13}=0$ no texture
zeros nor any such equalities can exist apart from the obvious ones. For
$\theta_{13}\ne 0$ some texture zeros become possible. In normal hierarchy two
texture zeros occur if $8.1\times 10^{-2}\leq |sin \theta_{13}|\leq 9.1\times
10^{-2}$ while in inverse hierarchy three are possible, one with $|sin
\theta_{13}|\geq 7\times 10^{-3}$ and two others with $|sin \theta_{13}|\geq
0.18$. All equalities between mass matrix elements are impossible with
$\theta_{13}\ne 0$.
