Someone asked a variant of this question recently (Equivalencity of $xI-A$). I ran across a suspiciously similar question today!
Let $A$ and $B$ be $n \times n$ matrices over $\mathbb{Q}$.
If $\det(xI-A) = \det(xI-B)$, can you say $xI-A$ and $xI-B$ are equivalent over $\mathbb{Q}[x]$? This is false and is answered here (Does equality of characteristic polynomials guarantee equivalence of matrices?)
If $xI-A$ and $xI-B$ are equivalent over $\mathbb{Q}[x]$, are $A$ and $B$ similar?
My suspicion is that (2.) is also false. If it were true, then we would have matrices $P,Q$ over $\mathbb{Q}[x]$ such that $xI-A = P(xI-B)Q$. And I am not sure how to reduce matrices over $\mathbb{Q}[x]$ to matrices over $\mathbb{Q}$ to get the required similarity relation. Although, this seems to be a slightly stronger statement than $A$ and $B$ have the same characteristic polynomial.
Any help appreciated.
