Write down the transfer matrix \(\boldsymbol{P}\) for a one-dimensional spin-1 Ising model in zero field, described by the Hamiltonian

\[
H_{N}\left\{\sigma_{i}\right\}=-J \sum_{i} \sigma_{i} \sigma_{i+1} \quad \sigma_{i}=-1,0,+1
\]

Show that the free energy of this model is given by

\[
\frac{1}{N} A(T)=-k T \ln \left\{\frac{1}{2}\left[(1+2 \cosh \beta J)+\sqrt{ }\left\{8+(2 \cosh \beta J-1)^{2}\right\}\right]\right\} .
\]

Examine the limiting behavior of this quantity in the limits \(T \rightarrow 0\) and \(T \rightarrow \infty\).