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integer matrix

Source: IMC 2001 day 1 problem 1

October 29, 2005
linear algebramatrixlinear algebra unsolved

Problem Statement

Let n n be a positive integer. Consider an n×n n\times n matrix with entries 1,2,...,n2 1,2,...,n^2 written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for n \equal{} 3 we get [123456789] \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]) We choose n n entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?
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