For every $\lambda \in \mathbf{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$. Prove that $J_n(\lambda)$ is invertible and determine in terms of $N$ and $\lambda$ the matrix $N'$ such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.
For every $\lambda \in \mathbf{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$.\\
Prove that $J_n(\lambda)$ is invertible and determine in terms of $N$ and $\lambda$ the matrix $N'$ such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.