Prove Invertibility of Matrix I + A for Projection Matrices A

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danik_ejik
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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.
 
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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.
 
If I+A and I-A/2 are supposed to be inverses then their product should be I, right? Is it?
 
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?
 
I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?
 
danik_ejik said:
I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?

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