41. In Prob. 40, let $B=C^{-1}$ and partition B as

$$B = \begin{bmatrix} B_{11} & b_{12} \\ b_{21} & b \end{bmatrix},$$

where $B_{11}$ is an $n \times n$ submatrix. Show that

(1) $b=0$,

(2) $b_{12}=(1/n)1$,

(3) $1'B_{11}=0$,

(4) $AB_{11}$ and $B_{11}A$ are idempotent.