Operations on a matrix
Source: Romania TST 1 2009, Problem 2
May 4, 2012
linear algebramatrixcombinatorics proposedcombinatorics
Problem Statement
Consider a matrix whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers , one can obtain, through a finite number of operations, a matrix having all entries divisible by . Prove that, through a finite number of operations, one can obtain the null matrix.