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Matrices that can be converted to 0 by certain operations

Source: 2012 China Mathematical Olympaid P2

January 13, 2012

linear algebramatrixgeometrygeometric transformationsymmetryalgebrapolynomial

Problem Statement

Let

p

be a prime. We arrange the numbers in

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

as a

p \times p

matrix

A = ( a_{ij} )

. Next we can select any row or column and add

1

to every number in it, or subtract

1

from every number in it. We call the arrangement good if we can change every number of the matrix to

0

in a finite number of such moves. How many good arrangements are there?