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Sum of B-entries equals the sum of corresponding A-entries

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November 21, 2010
linear algebramatrixcombinatorics proposedcombinatorics

Problem Statement

Given real numbers xi,yi(i=1,2,…,n)x_i,y_i (i=1,2,\ldots ,n), let AA be the n×nn\times n matrix given by aij=1a_{ij}=1 if xi≥yjx_i\ge y_j and aij=0a_{ij}=0 otherwise. Suppose BB is a n×nn\times n matrix whose entries are 00 and 11 such that the sum of entries in any row or column of BB equals the sum of entries in the corresponding row or column of AA. Prove that B=AB=A.
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