If $A^3=O$ is the matrice $A^2-A+I$ invertible? Is the matrice $A^2-A+I$ invertible? $A^2=A-I /\cdot A$ $ A^3 = A^2 - A$ $A^2=A$ Thus I conclude that $\lambda=0$ is one of the eigenvalues of this matrice and it isn't invertible, my conclusion comes from the fact that the eigenvalues of the matrices $A,A^2,...$ are all connected, and if $A^3=O$ that means that those two have the same eigenvalues.

$(A^2 - A + I)(A+I) = A^3 + I = I$.

To see how one knows that the matrix is invertible by just looking at it, one might recall the general fact:

> If $\sum_{n=0}^{\infty}(-1)^n A^n$ is convergent, then $I+A$ is invertible and one has:
>
> $$(I+A)^{-1} = \sum_{n=0}^{\infty}(-1)^n A^n$$

It's clearly the case here that the said series converges.