Proofs and matrices

Question

If A is a n x n matrix. Show that A^2 = 0, then if i - A is nonsingular and (i-A)^-1 = i + A

Thank you!

Answer 1

We can trivially check that if A^2=0 then (Id-A)(Id+A)=Id+A-A-A^2=Id and similarly, (Id+A)(Id-A)=Id. By definition the matrix Id-A is nonsingular since it has an inverse, namely the matric Id+A.