Prove that B cannot be obtain ed from A with some operations
Source:
September 22, 2010
linear algebramatrixIMO Shortlist
Problem Statement
Consider the two square matrices
A=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &-1&-1 &+1& +1 \\ +1 & -1 & -1 & -1 & +1 \\ +1 &+1&-1 &+1&-1 \end{bmatrix} \text{ and } B=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &+1&-1& +1&-1 \\ +1 &-1& -1& +1& +1 \\ +1 & -1& +1&-1 &+1 \end{bmatrix}with entries and . The following operations will be called elementary:(1) Changing signs of all numbers in one row;(2) Changing signs of all numbers in one column;(3) Interchanging two rows (two rows exchange their positions);(4) Interchanging two columns.Prove that the matrix cannot be obtained from the matrix using these operations.