IMC / 1994 / Problems / Day 1, P1

IMC 1994 · Day 1 · P1

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

Solution (official)

Denote by $a_{ij}$ and $b_{ij}$ the elements of $A$ and $A^{-1}$ , respectively. Then for $k \ne m$ we have $\sum\limits_{i=0}^{n} a_{ki} b_{im} = 0$ and from the positivity of $a_{ij}$ we conclude that at least one of $\{ b_{im} : i = 1,2,\dots,n \}$ is positive and at least one is negative. Hence we have at least two non-zero elements in every column of $A^{-1}$ . This proves part a). For part b) all $b_{ij}$ are zero except $b_{1,1} = 2$ , $b_{n,n} = (-1)^n$ , $b_{i,i+1} = b_{i+1,i} = (-1)^i$ for $i = 1,2,\dots,n-1$ .

IMC 1994 · Day 1 · P1

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