MathDB

6

Part of 2014 IMS

Problems(1)

n×n Matrix

Source: IMS 2014 - Day1 - Problem6

10/4/2014
Let A=[aij]n×nA=[a_{ij}]_{n \times n} be a n×nn \times n matrix whose elements are all numbers which belong to set {1,2,⋯ ,n}\{1,2,\cdots ,n\}. Prove that by swapping the columns of AA with each other we can produce matrix B=[bij]n×nB=[b_{ij}]_{n \times n} such that K(B)≤nK(B) \le n where K(B)K(B) is the number of elements of set {(i,j);bij=j}\{(i,j) ; b_{ij} =j\}.
linear algebramatrixlinear algebra unsolved