Linear Algebra - Given the Jordan form of $A \in Mat_7(\mathbb F)$, find Jordan form of $A^2+A+I_7$

Question

Given that the jordan form of the matrix $A \in Mat_7(\mathbb F)$ is:

$\begin{pmatrix} J_2(1) &\cdots &0\\0& \cdots J_3(1) \cdots &0\\0&0& \cdots J_2(2)\end{pmatrix}$

Find the Jordan form of $A^2+A+I$

Clarification: $J_{\alpha}(\beta)$ is a jordan block of size $\alpha$ with $\beta$ on the diagonal.

What I did:

I tried bruteforcing it, as in actually calculating $A^2+A+I$, but now to find the minimal polynomial of a 7x7 matrix...not fun. Is there a smarter way to go about this question?

From the jordan form of $A$ I can infer the minimal polynomial and characteristic polynomial of $A$. Does that have any implication on the characteristic and minimal polynomials of $A^2+A+I$?

Answer 1

Hint: If $$ A = SJS^{-1} $$ then

$$A^2+A+I = \left(SJS^{-1}\right)^2+\left(SJS^{-1}\right)+SS^{-1} = S\left(J^2+J+I\right)S^{-1}$$