Linear Algebra Inverse

Question

Let $$ be an $×$ matrix and $$ be the zero $×$ matrix.

a) Suppose that $^2=$. Prove that $+$ is invertible.

b) Suppose that $^=$ for some $$. Prove that $+$ is invertible.

I attached a picture of what I have done for part a but I think it's wrong and I have no idea what to do for part b)

Answer 1

If $A^2=0$, then $(\operatorname{Id}+A)(\operatorname{Id}-A)=\operatorname{Id}-A^2=\operatorname{Id}$, and therefore $(\operatorname{Id}+A)^{-1}=\operatorname{Id}-A$.

And if $A^k=0$,$$(\operatorname{Id}+A)(\operatorname{Id}-A+A^2-A^3+\cdots\pm A^{k-1})=\operatorname{Id}\pm A^k=\operatorname{Id},$$and therefore $(\operatorname{Id}+A)^{-1}=\operatorname{Id}-A+A^2-A^3+\cdots\pm A^{k-1}$.