Suppose there is a finite collection $\{P_k : k=1,2,...,K\}$ of transition matrices defined on the same finite state space $\Omega$. Each $P_k$ has entries $P_{k,i,j}$ for $i,j \in \Omega$. It is given that all $P_k$ are in detailed balance with respect to a common stationary distribution $\pi$ over $\Omega$. Show that $\frac{1}{K}\sum_{k=1}^{K} P_k$ and $\prod_{k=1}^{K} P_k$ are in global balance with respect to $\pi$.
My approach
I was trying to show for the mean, that $\pi_j =\sum_{i \in \Omega} \pi_i p^{*}_{ij}$ where $p^{*}_{ij}$ is the $(i,j)^{th}$ element of the mean transition matrix. This is equivalent to show that $$ \pi_j=\sum_{i \in \Omega} \pi _i \frac{1}{K} \sum_{k=1}^{K} p_{ijk} $$ $$ = \frac{1}{K} \sum_{i} \pi_i \sum_{k=1}^{K} \frac{\pi_j p_{jik}}{\pi_i} \quad \text{(from detailed balance)}$$ $$=\frac{1}{K} \sum_{k=1}^{K} \sum_{i \in \Omega} \pi_j p_{jik} = \frac{1}{K} \sum_{k=1}^{K} \pi_j = \pi_j$$
Is this correct? I can produce a similar argument for the product.