Showing that two real matrices are not congruent over $\mathbb{Q}$ Maybe it is a stupid question but I will still ask it here.
How can I prove that the following matrices are not congruent over $\mathbb{Q}$?
\begin{pmatrix}
  -1 & 0\\
  0 & 2\\
  \end{pmatrix}
\begin{pmatrix}
  -1 & 0\\
  0 & 1\\
  \end{pmatrix}
Thanks in advance

Fi $B=Q^TAQ$ then $\det(B)=\det(Q^T)\det(A)\det(Q)$, i.e. $\frac{\det(B)}{\det(A)}=\det^2(Q)$ must be a square.
