Question 12
(SWE 6) Consider the two square matrices
$$A=\begin{bmatrix}1&1&1&1&1\\1&1&1&-1&-1\\1&-1&-1&1&1\\1&-1&-1&-1&1\\1&1&-1&1&-1\end{bmatrix}\qquad B=\begin{bmatrix}1&1&1&1&1\\1&1&1&-1&-1\\1&1&-1&1&-1\\1&-1&-1&1&1\\1&-1&1&-1&1\end{bmatrix}$$
with entries $1$ and $-1$. 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 $B$ cannot be obtained from the matrix $A$ using these operations.
$$A=\begin{bmatrix}1&1&1&1&1\\1&1&1&-1&-1\\1&-1&-1&1&1\\1&-1&-1&-1&1\\1&1&-1&1&-1\end{bmatrix}\qquad B=\begin{bmatrix}1&1&1&1&1\\1&1&1&-1&-1\\1&1&-1&1&-1\\1&-1&-1&1&1\\1&-1&1&-1&1\end{bmatrix}$$
with entries $1$ and $-1$. 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 $B$ cannot be obtained from the matrix $A$ using these operations.