(Generalized Inverses) Let $$X^{-} = (X'X)^{-1}X'$$. Multiplied on the left, the matrix $$X^{-}$$ acts as an inverse matrix on X, providing a solution to the N equations

$$\mu = X\beta \iff (X'X)^{-1}X'\mu = (X'X)^{-1}X'X\beta = \beta$$

(a) What matrix do you get when you multiply X by $$X^{-}$$ on the right?

(b) Show that $X^- = (X'X)^{-1}X'$ has the defining property of a generalized inverse of a matrix:
$XX^-X = X$.

(c) Given an $N \times K$ matrix $Z$ such that $7'X = 0$, construct another generalized inverse for $X$ using $X^-$.

(d) Show that in addition $XX^-X^- = X^-$.

(e) Does your second generalized inverse have this additional property?

(f) Show that $X(X'X)^{-2}X'$ is a generalized inverse of $XX'$. Does it also possess the property described in Part d?
(g) Find a generalized inverse for $P_X$.