Solution
= Solution
<Detailed balance> means $\pi_i p_{ij}=\pi_jp_{ji}$ for all $i,j$. Summing over $i$ gives
$$
\sum_i\pi_i p_{ij}=\pi_j\sum_i p_{ji}=\pi_j,
$$
so \b[detailed balance implies invariance].
= Solution
<Detailed balance> means $\pi_i p_{ij}=\pi_jp_{ji}$ for all $i,j$. Summing over $i$ gives
$$
\sum_i\pi_i p_{ij}=\pi_j\sum_i p_{ji}=\pi_j,
$$
so \b[detailed balance implies invariance].