Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. We denote $\Pi_n = \begin{pmatrix} P(X_n = 0) \\ \vdots \\ P(X_n = N) \end{pmatrix} \in \mathcal{M}_{N+1,1}(\mathbb{R})$. Justify that, for all $n \in \mathbb{N}^*, \Pi_{n+1} = Q^\top \Pi_n$.
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. We denote $\Pi_n = \begin{pmatrix} P(X_n = 0) \\ \vdots \\ P(X_n = N) \end{pmatrix} \in \mathcal{M}_{N+1,1}(\mathbb{R})$. Justify that, for all $n \in \mathbb{N}^*, \Pi_{n+1} = Q^\top \Pi_n$.