What is the Determinant of an Idempotent Matrix?

[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 that's even the right step/method to take, or if its even right at all >_>

Thanks :)

 

[gabbagabbahey]

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

 

[forty]

det(P) = 1 or 0

 

[gabbagabbahey]

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

 

[gabbagabbahey]

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

 

[ahamdiheme]

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

 

[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

 

[kendarto]

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

 

[statdad]

kendarto: don't jump into another poster's thread.

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