TITLE: Prove that $A$ cannot be invertible if $A^2=0$
QUESTION [5 upvotes]: Let $A$ be an $n\times n$ matrix for which $A^2=0$. Prove that $A$ can not be invertible.

My attempt:
Given $A^2 = 0$, this means that $A = 0$. If $A$ is invertible, there must be an $n \times n$ matrix $B$ such that $AB = I$. However, because $A = 0$, this is not possible, thus $A$ is not invertible.

REPLY [3 votes]: If $A$ is invertible, then $A^2$ is invertible too. Thus $$I=(A^2)^{-1}A^2=(A^2)^{-1}0=0,$$ a contradiction.