Inverse matrix as a sum of matrix powers I have matrix $ A\in \mathbb{C}^{n x n}$ and $A$ is invertible.
How can I show that coefficients $c_0,...,c_{n-1}$ exist :
$A^{-1} = c_0I+c_1A+...+c_{n-1}A^{n-1}$
I tried to solve it first by multiplying the equation above with $A$ which gives:
$I = c_0A + c_1A^2 + ... +c_{n-1}A^n$
And I continued then by knowing that every square matrix can be written with Jordan canonical form ($A = VJ_aV^-1$):
$I = c_0VJ_aV^{-1} + c_1VJ_a^2V^{-1}+...+c_{n-1}VJ_a^nV^{-1}$
Which can be written in form:
$I = V(c_0J_a+c_1J_a^2+....+c_{n-1}J_a^n)V^{-1}$
After this point my task is to show that the identity matrix can be presented with sum of coefficients and Jordan canonical forms, but I don't have any idea how to do that or even if my approach to this problem is right.

Hint
The constant coefficient of the characteristic polynomial is $\det(A)$ and by the Cayely-Hamilton theorem it annihilates $A$...
