  We characterise all linear maps $\mathcal{A}\colon\mathbb{R}^{n\times
n}\to\mathbb{R}^{n\times n}$ such that, for $1\leq p<n$, \begin{align*}
\|P\|_{L^{p^{*}}(\mathbb{R}^{n})}\leq
c\,\Big(\|\mathcal{A}[P]\|_{L^{p^{*}}(\mathbb{R}^{n})}+\|\mathrm{Curl}
P\|_{L^{p}(\mathbb{R}^{n})} \Big) \end{align*} holds for all compactly
supported $P\in C_{c}^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n\times n})$, where
$\mathrm{Curl} P$ displays the matrix curl. Being applicable to incompatible,
that is, non-gradient matrix fields as well, such inequalities generalise the
usual Korn-type inequalities used e.g. in linear elasticity. Different from
previous contributions, the results gathered in this paper are applicable to
all dimensions and optimal. This particularly necessitates the distinction of
different constellations between the ellipticities of $\mathcal{A}$, the
integrability $p$ and the underlying space dimensions $n$, especially requiring
a finer analysis in the two-dimensional situation.
