Question

Show that if A^2 =0 then I-A in invertible and (I-A)^-1=I+A.

Answer 1

Let A be an nxn matrix. Show that if A^(2) = 0, then I-A is nonsingular and (I-A)^(-1) = I+A

Note that (I - A)(I + A)
= I(I + A) - A(I + A)
= (I - A) - (A + A^2)
= I - A^2
= I - 0, since A^2 = 0
= I.

Hence, I - A is non singular with inverse I + A (since the inverse is unique when it does exist)