Consider simple linear regression with $n$ pairs of numbers $x_i,y_i$. Consider the $n\times 2$ matrix ${\bf X}$ with the first column consisting of 1s and the second column consisting of the $x_i$. Below are some questions on the resulting $({\bf X}^T{\bf X})$ and $({\bf X}^T{\bf X})^{-1}$.
To do these questions, please review the results for the determinant and inverse of a 2x2 matrix.

Part a)
Which expressions match the determinant of ${\bf X}^T{\bf X}$? Possibly more than one item is correct.

A. $\sum_i (x_i-{\bar x})^2$
B. $n s_x^2$
C. $n(n-1)s_x^2$
D. $n\sum_i (x_i-{\bar x})^2$
E. None of the above


Part b)
Let det denote the determinant of $({\bf X}^T{\bf X})$.
Which expression matches the (2,2) element of $({\bf X}^T{\bf X})^{-1}$?

A. $-\sum_i x_i^2$/det
B. $\sum_i x_i^2$/det
C. $n$/det
D. $-n$/det
E. 1/det
F. None of the above


Part c)
Which expression matches the (1,2) element of $({\bf X}^T{\bf X})^{-1}$?

A. $\sum_i x_i$/det
B. $n$/det
C. $-n$/det
D. $-\sum_i x_i$/det
E. 1/det
F. None of the above


Part d)
In simple linear regression, the standard error of an estimate of the subpopulation mean at $x^*$ is For multiple regression, the standard error of an estimate of the subpopulation mean at ${\bf x}^*=(1,x_1^*,\ldots,x^*_p)^T$ is When $p=1$ and ${\bf x}^*=(1,x^*)^T$, the two equations should be the same. Check this by simplifying (The above expression means the determinant multiplied by remainder). Which are following are equivalent to the equation (A3)?
Possibly more than one item is correct.

A. $n(n-1)s_x^2$
B. $\sum_i x_i^2 +2x^*\sum_i x_i + n(x^*)^2$
C. $\sum_i x_i^2 +2nx^*{\bar x} + n(x^*)^2$
D. $(n-1)s_x^2 + n{\bar x}^2 -2nx^*{\bar x} + n(x^*)^2$
E. $\sum_i x_i^2 -2x^*\sum_i x_i + n(x^*)^2$
F. $(n-1)s_x^2 + n{\bar x}^2 +2nx^*{\bar x} + n(x^*)^2$
G. $(n-1)s_x^2 + n(x^*+{\bar x})^2$
H. $(n-1)s_x^2 + n(x^*-{\bar x})^2$
I. $\sum_i x_i^2 -2nx^*{\bar x} + n(x^*)^2$
J. None of the above


Part e)
After doing (d), you should be able to complete the derivation of the equivalence of (A1) and (A2) for $p=1$.

Hint: