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Minimal and maximal possible rank of a matrix

Source: IMC 2007, Day 1, Problem 2

August 6, 2007

linear algebramatrixinductionIMCcollege contests

Problem Statement

Let

n\ge 2

be an integer. What is the minimal and maximal possible rank of an

n\times n

matrix whose

n^{2}

entries are precisely the numbers

1, 2, \ldots, n^{2}