[Math] Showing that two real matrices are not congruent over $\mathbb{Q}$

Question

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

Answer 1

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.