  For the unitary ensembles of $N\times N$ Hermitian matrices associated with a
weight function $w$ there is a kernel, expressible in terms of the polynomials
orthogonal with respect to the weight function, which plays an important role.
For the orthogonal and symplectic ensembles of Hermitian matrices there are
$2\times2$ matrix kernels, usually constructed using skew-orthogonal
polynomials, which play an analogous role. These matrix kernels are determined
by their upper left-hand entries. We derive formulas expressing these entries
in terms of the scalar kernel for the corresponding unitary ensembles. We also
show that whenever $w'/w$ is a rational function the entries are equal to the
scalar kernel plus some extra terms whose number equals the order of $w'/w$.
General formulas are obtained for these extra terms. We do not use
skew-orthogonal polynomials in the derivations.
