  We construct a collection of matrices defined by quadratic residue symbols,
termed "quadratic residue matrices", associated to the splitting behavior of
prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and prove
a simple criterion characterizing such matrices. We also study the analogous
classes of matrices constructed from the cubic and quartic residue symbols for
a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$,
respectively.
