IMC 1994 Problem 1

a) Let $A$ be a $n\times n$ , $n\geq 2$ , symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$ , where $z_n$ is the number of zero elements in $A^{-1}$ .

b) How many zero elements are there in the inverse of the $n\times n$ matrix $$A=\begin{pmatrix} 1&1&1&1&\ldots&1 1&2&2&2&\ldots&2 1&2&1&1&\ldots&1 1&2&1&2&\ldots&2 \vdots&\vdots&\vdots&\vdots&\ddots&\vdots 1&2&1&2&\ldots&\ddots \end{pmatrix}$$