MathDB

7

Part of 1984 Austrian-Polish Competition

Problems(1)

permutation of numbers on a matrix so that \sum b_{ij} < 2

Source: Austrian Polish 1984 APMC

4/30/2020
A m×nm\times n matrix (aij)(a_{ij}) of real numbers satisfies ∣aij∣<1|a_{ij}| <1 and ∑i=1maij=0\sum_{i=1}^m a_{ij}= 0 for allj j. Show that one can permute the entries in each column in such a way that the obtained matrix (bij)(b_{ij}) satisfies ∑j=1nbij<2\sum_{j=1}^n b_{ij} < 2 for all ii.
linear algebramatrix