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### Symbolic matrices and "integrity" of their inverse

I have to solve the following problem:

Does a matrix $G\in GL(n,\mathbb{Z})$ exists such that $$G\times A\times G^{-1}=B$$ being $A,B$ given matrices in $\mathbb{Q}$?

Doing everything by hand, I finally find myself with a bunch of symbolic matrices. Now I have to check if they can lay inside $GL(n,\mathbb{Z})$, i.e. if there are integer values for the variables in the matrix such that the matrix is integer, invertible and with integer inverse.

E.g.: $$\left(\begin{array}{cc}x & 0 \ 0 & y\end{array}\right)$$ does the trick only for $x=y=1$.

Is there a quick method within Sage to solve that last problem?

Thanks!

### Symbolic matrices and "integrity" of their inverse

I have to solve the following problem:

Does a matrix $G\in GL(n,\mathbb{Z})$ exists such that $$G\times A\times G^{-1}=B$$ being $A,B$ given matrices in $\mathbb{Q}$?

Doing everything by hand, I finally find myself with a bunch of symbolic matrices. Now I have to check if they can lay inside $GL(n,\mathbb{Z})$, i.e. if there are integer values for the variables in the matrix such that the matrix is integer, invertible and with integer inverse.

E.g.: $$\left(\begin{array}{cc}x & 0 \ \\ 0 & y\end{array}\right)$$ does the trick only for $x=y=1$.

Is there a quick method within Sage to solve that last problem?

Thanks!