Help with proof or counterexample: $A^3=0 \implies I_n+A$ is invertible

Question

I'm having a bit of trouble with the following proof, and I was hoping someone could help. I started with some things already but I'm not sure where to go from there.

Suppose $A\in M_{n\times n}(\mathbb{R})$ for $n\geq 2$ and let $I_n$ be the $n\times n$ identity matrix. Prove, or give a counter example, that if $A^3=0$, then $I_n+A$ is invertible.

Proof:

We know that $A^3=0$ and $det(0)=0$ such that $det(A^3)=det(A)\times det(A)\times det(A)=0$. From that it follows that $det(A)=0$.

We want to prove that $I_n+A$ is invertible, so $det(I_n+A)$ must be unequal to $0$.

We know that $det(I_n+A)=\sum_{j=1}^n\Big[(-1)^{j+1}(I_n+A)_{1j}\times det\widetilde{(I_n+A)}_{1j}\Big]$ with the tilde determinant representing the minor, deleting row $i$ and $j$... I'm not sure how to go from here, anyone?

Answer 1

Because just $$(I+A)(I-A+A^2)=I+A^3=I$$

Answer 2

If $A^3=0$, its only eigenvalue is $0$, therefore the only eigenvalue of $A+I$ is $1$. The determinant of a matrix is equal to the product of its eigenvalues, so $\det(A+I)\ne0$.