Projection $(I-p)=(I+p)$

Question

Let $p$ be a projection such that the projection $(I-p)$ is invertible. Does $(I-p)=(I+p)$ hold?

I proved it this way:

$$(I-p)^2=(I-p)=(I^2-p^2)=(I-p)(I+p)$$

Hence,

$$(I-p)^{-1}(I-p)^2= (I+p).$$

This is equivalent to

$$(I-p)=(I+p).$$

I used properties of projections $p^2=p$ and $(I-p)^2=(I-p)$.

Answer 1

Your proof is correct, but the result isn't very interesting. In fact, the only invertible projection $P'$ is $P' = I$ since $P' = (P')^{-1}(P')^2 = (P')^{-1}P' = I$. Thus $I - P$ invertible means $I - P = I$, i.e. $P = 0$.