Are Trace of product of matrices- distributive/associative? Is $\operatorname{Tr}(X^TAX)-\operatorname{Tr}(X^TBX)$ equal to $\operatorname{Tr}(X^TCX)$, where $C=A-B$ and $A$, $B$, $X$ have real entries and also $A$ and $B$ are p.s.d. 

Yes, as 
$$X^t(A-B)X=X^t(AX-BX)=X^tAX-X^tBX,$$
using associativity and distributivity of product with respect to the addition. The fact that the matrices $A$ and $B$ are p.s.d. is not needed here.
