How to prove idempotent and find the rank

Question

How to solve these three parts

Let X be a n×p matrix such that X'X has an inverse. Let A = X(X'X)^-1X'.
a) Show that A is idempotent.
b) Show that P = In - A is also idempotent.
c) Find the rank of P.

Thanks :)

Answer 1

I suppose that $X'$ means $X^T$.

a) You have to show that $A^2=A$. But this is straightforward: compute

$A^2=(X(X^TX)^{-1}X^T)(X(X^TX)^{-1}X^T)=.....= X(X^TX)^{-1}X^T=A$.

b) $P^2=(I_n-A)^2=I_n^2-2A+A^2=I_n-2A+A=I_n-A=P$.

It is your turn to compute the rank of $P$. Hint: $im(P)=ker(I_n-P)=ker(A)$.