  The inverse of the metric matrices on the Siegel-Jacobi upper half space
${\mathcal{X}}^J_n$, invariant to the restricted real Jacobi group
$G^J_n(\mathbb{R})_0$ and extended Siegel-Jacobi $\tilde{{\mathcal{X}}}^J_n$
upper half space, invariant to the action of the real Jacobi
$G^J_n(\mathbb{R})$, are presented. The results are relevant for Berezin
quantization of the manifolds ${\mathcal{X}}^J_ n$ and
$\tilde{\mathcal{X}}^J_n$. Explicit calculations in the case $n=2$ are given.
