Chapter 5.3, Problem 11E

Let $P=A{\left(A^{T}A\right)}^{-1}{A}^T,$ where A is an $m\times n$ matrixof rank n.
(a)Show that $P^2=P.$
(b) Provethat $P^k=P$ for $k=1,2,....$
(c)Show that P is symmetric. [Hint: If B is nonsingular, then ${\left(B^{-1}\right)}^T={\left(B^T\right)}^{-1}.]$