Matrix Proof: Idempotent

1. Oct 7, 2008

forty

A matrix P is called idempotent if P^2 = P. If P is idempotent and P =/= I show that det(P)=0.

I don't really know where to go with this but i have a feeling that it involves taking the det of each side.

det(P^2) = det(P)
det(P)det(P) = det(P)

where to from here if thats even the right step/method to take, or if its even right at all >_>

Thanks :)

2. Oct 7, 2008

gabbagabbahey

looks fine to me; now for what values of det(P) does that equation hold?

3. Oct 7, 2008

forty

det(P) = 1 or 0

4. Oct 7, 2008

gabbagabbahey

Okay, if detP=1, and P^2=P, what matrix must P be?

Last edited: Oct 7, 2008
5. Oct 7, 2008

gabbagabbahey

Hint: use the fact that if $det(P) \neq 0$, then P is invertible. Multiply $P^2=P$ by $P^{-1}$.

6. Oct 27, 2008

ahamdiheme

but it says det(P)=/=1. How do you show that det(P)=0???

7. Oct 27, 2008

rock.freak667

det(P2) = det(P)

=> det(P)^2-det(P)=0

This is the same as t^2-t=0 where t=det(P). Factorise and use that fact that P=/= I

8. Sep 13, 2009

kendarto

I have a questions:
if A=I-X(X'X)^-1X'
is it A idempotent?

9. Sep 13, 2009

try to calculate $$A^2$$ and answer this for yourself.