46. In Prob. 45 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.