IMO Shortlist 1973 problem 12
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Avg: 0,0 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}\quad\text{ and }\quad 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}](/media/m/1/4/1/14140ea6705664d7d27f548c3a8e910c.png)
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.
![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}\quad\text{ and }\quad 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}](/media/m/1/4/1/14140ea6705664d7d27f548c3a8e910c.png)
with entries
![+1](/media/m/5/2/e/52e8a85276568847ceb91e7eeb5865e5.png)
![-1](/media/m/6/1/c/61cf05f5d8d6a4f0d373e7452cde9c3c.png)
(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
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1973