  We investigate and discuss when the inverse of a multivariate truncated
moment matrix of a measure $\mu$ has zeros in some prescribed entries. We
describe precisely which pattern of these zeroes corresponds to independence,
namely, the measure having a product structure. A more refined finding is that
the key factor forcing a zero entry in this inverse matrix is a certain
conditional triangularity property of the orthogonal polynomials associated
with $\mu$.
