Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-d\not\equiv 1 [4]$.
For all $a,b,c\in\mathbb{Z}$ such that $ac-b^2=d,$ set $[a,b,c]_d=\begin{pmatrix}b & -c \cr a & -b\end{pmatrix}\in M_2(\mathbb{Z})$.
(Note that the relation $ac-b^2=d$ forces $a\neq 0$ and $c\neq 0$.)
Let's say that $[a,b,c]_d$ is reduced if $0<a\leq c, \ -\dfrac{a}{2}<b\leq \dfrac{a}{2}$, and $b\geq 0$ whenever $a=c$.
Note the similarity with the notion of reduced positive binary form à la Gauss (up to a factor $1/2$ which might be easily explained); this is not a coincidence.
Question. Without using the theory of binary forms, is it possible to prove that, if $[a,b,c]_d$ and $[a',b,'c']_d$ are reduced and similar over $\mathbb{Z}$ (that is, conjugate by an element of $GL_2(\mathbb{Z})$), then $a=a',b=b',c=c'$.
[Let me explain why it should be true. There is a bijection between the set of conjugacy classes of integral matrices with characteristic polynomial $X^2+d$ and the set of equivalences of ideals of $\mathbb{Z}[\sqrt{-d}]$, where two ideals $I$ and $J$ are equivalent if there exist $z,z'\in \mathbb{Q}(\sqrt{-d})$ such that $zI=z'J$.
But if I am not mistaken, it is known that the set of classes of ideals is in bijection with the set of equivalent positive definite binary forms of discriminant $-4d$. And such a form is properly equivalent to a unique reduced binary form. Now, my matrix $[a,b,c]_d$ is reduced if and only if $ax^2+2bxy+cy^2$ is reduced.]
Thanks in advance for your help!
G.
