For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$. Show that, for all $k \in \llbracket 1,n \rrbracket$, $C_{n,1}^k = C_{n,k}$.
For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by
$$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$
We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Show that, for all $k \in \llbracket 1,n \rrbracket$, $C_{n,1}^k = C_{n,k}$.