Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$ and $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N} \in \mathcal{M}_{N+1}(\mathbb{R})$. Justify that $A(t)$ possesses a dominant eigenvalue $\gamma(t) > 0$.