  The $j$-multiplicity plays an important role in the intersection theory of
St\"uckrad-Vogel cycles, while recent developments confirm the connections
between the $\epsilon$-multiplicity and equisingularity theory. In this paper
we establish, under some constraints, a relationship between the
$j$-multiplicity of an ideal and the degree of its fiber cone. As a
consequence, we are able to compute the $j$-multiplicity of all the ideals
defining rational normal scrolls. By using the standard monomial theory, we can
also compute the $j$- and $\epsilon$-multiplicity of ideals defining
determinantal varieties: The found quantities are integrals which, quite
surprisingly, are central in random matrix theory.
