Does there exist a non-singular matrix $Q \in \mathbb Q^{n\times n}$ with the following two properties?

Question

I am trying to find a non-singular matrix $Q \in \mathbb Q^{n\times n}$ such that:

1. The characteristic polynomials $char(Q)$ and $char(Q^{-1})$ both have some non-integer coefficients.
2. The $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$ does not contain $\mathbb Z^n$, i.e $Span_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\mathbb Z^{n})\} \nsupseteq\mathbb Z^n$.

It's quite easy to find a matrix that satisfies (1), e.g. $Q=\begin{bmatrix} 1.5 & 1 \\ 1 & 0 \end{bmatrix}.$ However, I can't find a matrix that satisfies both properties.

Any ideas to construct such a matrix would be really appreciated.

Answer 1

Thanks to Jeremy, here is an example:

$Q=\begin{bmatrix} 1/3 & 0 & 0 \\ 1/2 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$