  Let $\mathcal{A}$ be a connected cochain DG algebra, whose underlying graded
algebra $\mathcal{A}^{\#}$ is the quantum affine $n$-space
$\mathcal{O}_{-1}(k^n)$. We compute all possible differential structures of
$\mathcal{A}$ and show that there exists a one-to-one correspondence between
$$\{\text{cochain DG
algebra}\,\,\mathcal{A}\,|\,\mathcal{A}^{\#}=\mathcal{O}_{-1}(k^n)\}$$ and the
$n\times n$ matrices $M_n(k)$. For any $M\in M_n(k)$, we write
$\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ for the DG algebra corresponding to
it. We also study the isomorphism problems of these non-commutative DG
algebras. For the cases $n\le 3$, we check their homological properties. Unlike
the case of $n=2$, we discover that not all of them are Calabi-Yau when $n=3$.
In spite of this, we recognize those Calabi-Yau ones case by case. In brief, we
solve the problem on how to judge whether a given such DG algebra
$\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ is Calabi-Yau.
