grandes-ecoles 2018 Q35

grandes-ecoles · France · centrale-maths1__psi Matrices Linear System and Inverse Existence

Let $C$ be a nilpotent matrix. Show that $I_n + C$ is invertible and that $$\left(I_n + C\right)^{-1} = I_n - C + C^2 + \cdots + (-1)^{n-1} C^{n-1}$$

Let $C$ be a nilpotent matrix. Show that $I_n + C$ is invertible and that
$$\left(I_n + C\right)^{-1} = I_n - C + C^2 + \cdots + (-1)^{n-1} C^{n-1}$$