  We give a necessary and a sufficient condition for the boundedness of the
Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman space
$L_a^2(\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with scalar valued
Bergman space entries. The results generalize existing results in the scalar
valued Bergman space case. We also characterize boundedness and invertibility
of Toeplitz products $T_FT_{G^*}$ in terms of the Berezin transform,
generalizing results found by Zheng and Stroethoff for the scalar valued
Bergman space.
