  We prove a local $Tb$ theorem for paraproducts acting on vector valued
functions, with matrix weighted averaging operators. The condition on the
weight is that its square is in the $L_2$ associated matrix $A_\infty$ class.
We also introduce and use a new matrix reverse H\"older class. This result
generalizes the previously known case of scalar weights from the proof of the
Kato square root problem, as well as the case of diagonal weights, recently
used in the study of boundary value problems for degenerate elliptic equations.
