IMC 1994 · Problem 1

a) Let $A$ be a $n \times n$, $n \ge 2$, symmetric, invertible matrix with real positive elements. Show that $z_n \le n^2 - 2n$, 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

$$A = \begin{pmatrix} 1 & 1 & 1 & 1 & \dots & 1 \\ 1 & 2 & 2 & 2 & \dots & 2 \\ 1 & 2 & 1 & 1 & \dots & 1 \\ 1 & 2 & 1 & 2 & \dots & 2 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 2 & 1 & 2 & \dots & \dots \end{pmatrix}?$$