  We consider four nontrivial ensembles involving Gaussian Wigner and Wishart
matrices. These are relevant to problems ranging from multiantenna
communication to random supergravity. We derive the matrix probability density,
as well as the eigenvalue densities for these ensembles. In all cases the joint
eigenvalue density exhibits a biorthogonal structure. A determinantal
representation, based on a generalization of Andr\'{e}ief's integration
formula, is used to compactly express the $r$-point correlation function of
eigenvalues. This representation circumvents the complications encountered in
the usual approaches, and the answer is obtained immediately by examining the
joint density of eigenvalues. We validate our analytical results using Monte
Carlo simulations.
