# Homework Help: Matrix Proof Help

1. Sep 24, 2010

### danik_ejik

Hello,
I need a hint on how to begin this proof, please.

Prove that if A is a projection matrix, A2=A, then I + A is invertible and
(I + A) -1 = I - $$\frac{1}{2}$$A.

2. Sep 24, 2010

### hunt_mat

I would write B=a*I+b*A and compute B*(I+A) and (I+A)B and see for what values of a and b I find an inverse. Note that A^2=A, so there are no higher powers in the series expansion of the inverse.

3. Sep 24, 2010

### Dick

If I+A and I-A/2 are supposed to be inverses then their product should be I, right? Is it?

4. Sep 24, 2010

Well, $$I-A/2$$ is a matrix. Why don't you try to see what you get if you multiply it by $$I+A$$? Maybe it will tell you something?

5. Sep 24, 2010

### danik_ejik

I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?

6. Sep 24, 2010

### Dick

Sure that proves it. If MN=I then M=N^(-1) and N=M^(-1). It's the definition of 'inverse'.

7. Sep 24, 2010

thank you