Given the matrix $A$ such that $A^3=0$. Determine if matrix $A^2-A+I$ is invertible, and if it is, find it's inverse.

Question

Given the matrix $A$ such that $A^3=0$. Determine if matrix $A^2-A+I$ is invertible, and if it is, find it's inverse.

Is it possible to directly evaluate matrix $A$ from the condition $A^3=0$? If not, can we use Cayley-Hamilton theorem?

Answer 1

($A+I$)($A^2$-$A$+$I$)=$I$

So that is invertible.