TITLE: Smallest possible rank of an $n×n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal
QUESTION [6 upvotes]: A problem in IMC 2012 in which i'm interested but I have no answer. Can you help me? Many thanks.
Problem : Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n\times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal.

REPLY [3 votes]: The problem has been solved at the website Art Of Problem Solving. The minimal rank is $2$ for two times two matrices, and $3$ for $n\geq 3$. A matrix of minimal rank is given: its entries are $a_{i,j}:=(i-j)^2$.