Prove that if $A^2=0$ then $A$ is not invertible Let $A$ be $n\times n$ matrix. Prove that if $A^2=\mathbf{0}$ then $A$ is not invertible.

Assume there is an inverse denoted A-1. Then you have AA = 0 and then multiply it by your inverse A-1 you have AAA-1 = 0A-1 = A = 0.
Now given the definition of 0 :
Given a Matrix M and the0vector M0=0=0M.
Therefore by definition of0it cannot have an inverse.
