Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$ for all $k \in \mathbb{N}$. Prove that $\forall k \in \mathbb { N } , \Phi _ { k + p } = \Phi _ { k } \Phi _ { p }$.
Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$ for all $k \in \mathbb{N}$. Prove that $\forall k \in \mathbb { N } , \Phi _ { k + p } = \Phi _ { k } \Phi _ { p }$.