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 B₁₁ is an *n x n* submatrix. Show that

(1) b = 0,

(2) b₁₂ = (1/n)1,

(3) 1'B₁₁ = 0,

(4) AB₁₁ and B₁₁A are idempotent.