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@JoséCarlosSantos Would this be satisfactory: If $A$ is an $n \times n$ matrix such that $A^2=0$, is $A+I_{n}$ invertible?

I have also asked in the searching chatroom: chat.stackexchange.com/transcript/46148/2020/1/14 - maybe somebody will find other duplicates or similar questions.

There is more of them: math.stackexchange.com/questions/linked/19538

There is also this generalization: Prove that $A+I$ is invertible if $A$ is nilpotent

@MartinSleziak The first one is good enough. Thanks a lot.

duplicates These questions seem to be tied to the same topic brought up by José Carlos Santos.

19538: If $A$ is an $n \times n$ matrix such that $A^2=0$, is $A+I_{n}$ invertible?

1439244: Proving the invertibility of an $n$ by $n$ matrix (Linear algebra)

2261978: Say $A^2 = 0$. Prove that for every real number $a$, the matrix $I + a A$ is invertible

2329604: $I-A$ and $I+A$ are invertible when $A^2=0$

3508783: If $A^2$ of square matrix $A$ is the null matrix, then $A + I$ is invertible

It's quite plausible that there are more similar questions.

I am not sure which is the best duplicate target - to some extent, the oldest one seems a bit like solution-verification question.

Delete: DA, DB, DC, DD, DE, DF, DG, DH

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@JoséCarlosSantos For dupes, we should strive to do better than "good enough", i.e. if the search is not so difficult then we should strive to steer students to the "best" answers. Think back to when you were a student, would you prefer a teacher who gave you "good enough" instruction or "best"?

@MartinSleziak As always, your efforts to help find the best dupes are much appreciated.

D1, D2, D3.

D4, D5, D6.

D7, D8, D9.

D10, D11, D12.

D13, D14, D15.