Proving Idempotency and Nonsingularity of Matrix A

[Roni1985]

Homework Statement

Let A be an idempotent matrix (A^2=A)
a) show that I-A is also idempotent
b) show that I+A is nonsingular and (I+A)^(-1)=I-(1/2)*A

Homework Equations

The Attempt at a Solution

I managed to solve part a, but have a little problem solving b.

I tried
(I+A)*(I+A)^(-1)=(?)I

but didn't know how to treat (I+A)^(-1)

Thanks,
Roni.

 

[Mark44]

You are given that A² = A.
Adding I to both sides gives us
A² + I = A + I
==> |A² + I| = |A + I|

Can you do anything with that?

 

[Roni1985]

You are given that A² = A.
Adding I to both sides gives us
A² + I = A + I
==> |A² + I| = |A + I|

Can you do anything with that?

oh, nice approach.
Thanks for the help.
I'll try the second part of B