MathDB
Show that all indices are zero except four

Source: Iran Third Round MO 1998, Exam 1, P3

July 1, 2012
linear algebramatrixlinear algebra unsolved

Problem Statement

Let A,BA,B be two matrices with positive integer entries such that sum of entries of a row in AA is equal to sum of entries of the same row in BB and sum of entries of a column in AA is equal to sum of entries of the same column in BB. Show that there exists a sequence of matrices A1,A2,A3,⋯ ,AnA_1,A_2,A_3,\cdots , A_n such that all entries of the matrix AiA_i are positive integers and in the sequence A=A0,A1,A2,A3,⋯ ,An=B,A=A_0,A_1,A_2,A_3,\cdots , A_n=B, for each index ii, there exist indexes k,j,m,nk,j,m,n such that \begin{array}{*{20}{c}} \\ {{A_{i + 1}} - {A_{i}} = } \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}}     \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { + 1}&{ - 1} \\ { - 1}&{ + 1} \end{array}} \right)} \end{array} \ \text{or} \ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}}     \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { - 1}&{ + 1} \\ { + 1}&{ - 1} \end{array}} \right)} \end{array}. That is, all indices of Ai+1−Ai{A_{i + 1}} - {A_{i}} are zero, except the indices (m,j),(m,k),(n,j)(m,j), (m,k), (n,j), and (n,k)(n,k).
Back to ProblemsView on AoPS