Proof that a matrix is nonsingular Let $A$ be $n \times n$ matrix. Show that if $A^2 = 0$, then $I - A$ is non-singular and $(I-A)^{-1} = I+A$.
The second part is easy for me, but how can I show that if $A^2 = 0$, then $I - A$ is non-singular. I found in Wolfram Alpha that "A matrix is singular iff its determinant is 0.", but how I can relate this to the given $A^2 = 0$. or is there another easier way. Thank you for your help in advance.

Comparing $|I-A|$ to the characteristic polynomial of $A$:
$|I-A| = 0 \implies \lambda=1$ is an eigenvalue of $A$. But $A$ nilpotent necessitates that the only eigenvalue of $A$ is $0$, contradiction.
