Is the average of positive-definite matrices also positive-definite?

Yes, it is. jth asnwer is correct (+1) but I think you can get a much simple explanation with just basic Linear Algebra.

Assume $A$ and $B$ are positive definite matrices for size $n$. By definition this means that for all $u \in R^n$, $0 < u^TAu$ and $0 < u^TBu$.
This means that $0 < u^TAu + u^TBu$ or equivalently that $0 < u^T(A+B)u$. ie. $(A+B)$ has to be positive definite.