  In this article we examine the densities of a product and a ratio of two real
positive definite matrix-variate random variables $X_1$ and $X_2$, which are
statistically independently distributed, and we consider the density of the
product $U_1=X_2^{1\over2}X_1X_2^{1\over2}$ as well as the density of the ratio
$U_2=X_2^{1\over2}X_1^{-1}X_2^{1\over2}$. We define matrix-variate Kober
fractional integral operators of the first and second kinds from a statistical
perspective, making use of the derivation in the predecessor of this paper for
the scalar variable case, by deriving the densities of products and ratios
where one variable has a matrix-variate type-1 beta density and the other
variable has an arbitrary density. Various types of generalizations are
considered, by using pathway models, by appending matrix variate hypergeometric
series etc. During this process matrix-variate Saigo operator and other
operators are also defined and properties studied.
