Your Textbook Shows That the Following Matrix (M_x = I_n $\frac { 1 } { n }$

question 16

Essay

Your textbook shows that the following matrix (M_x = I_n - P_x)is a symmetric idempotent matrix. Consider a different Matrix A, which is defined as follows: A = I - $\frac { 1 } { n }$ ιι' and ι = $\left[ \begin{array} { c } 1 \\1 \\\ldots \\1\end{array} \right]$ a. Show what the elements of A look like.
b. Show that A is a symmetric idempotent matrix
c. Show that Aι = 0.
d. Show that A $\hat { \mathrm { U } }$ = $\hat { \mathrm { U } }$ , where $\hat { \mathrm { U } }$ is the vector of OLS residuals from a multiple regression.