  We establish a weighted inequality for the Bergman projection with matrix
weights for a class of pseudoconvex domains. We extend a result of
Aleman-Constantin and obtain the following estimate for the weighted norm of
$P$: \[\|P\|_{L^2(\Omega,W)}\leq C(\mathcal B_2(W))^{{2}}.\] Here $\mathcal
B_2(W)$ is the Bekoll\'e-Bonami constant for the matrix weight $W$ and $C$ is a
constant that is independent of the weight $W$ but depends upon the dimension
and the domain.
