We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Deduce that (II.2) admits a nonzero periodic solution of period $p$ if and only if $\operatorname { tr } ( Q ) = 2$. Prove that in this case, either all solutions of (II.2) are periodic of period $p$, or (II.2) admits an unbounded solution. One may prove that there exists a matrix $P \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ and a complex number $\alpha$ such that $Q = P \left( \begin{array} { c c } 1 & \alpha \\ 0 & 1 \end{array} \right) P ^ { - 1 }$ and, in the case where $\alpha \neq 0$, consider the solution of Sol(II.2) whose image by $\Psi$ is the vector $P \binom { 0 } { 1 }$.
We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Deduce that (II.2) admits a nonzero periodic solution of period $p$ if and only if $\operatorname { tr } ( Q ) = 2$. Prove that in this case, either all solutions of (II.2) are periodic of period $p$, or (II.2) admits an unbounded solution.
One may prove that there exists a matrix $P \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ and a complex number $\alpha$ such that $Q = P \left( \begin{array} { c c } 1 & \alpha \\ 0 & 1 \end{array} \right) P ^ { - 1 }$ and, in the case where $\alpha \neq 0$, consider the solution of Sol(II.2) whose image by $\Psi$ is the vector $P \binom { 0 } { 1 }$.