  Boundedness for a class of projection operators, which includes the
coordinate projections, on matrix weighted $L^p$-spaces is completely
characterised in terms of simple scalar conditions. Using the projection
result, sufficient conditions, which are straightforward to verify, are
obtained that ensure that a given matrix weight is contained in the Muckenhoupt
matrix $A_p$ class. Applications to singular integral operators with product
kernels are considered.
