for any matrix $M=[m_{ij}]$.

$(a)$ Since $\T$ is invertible, the semi-linear map $\T$ is injective. Hence $|\mathscr{C}\star \T |=|\mathscr{C}|$. From $\mathscr{C}\star \T\subseteq \mathscr{C}$ we deduce that $\mathscr{C}\star \T=\mathscr{C}$.

$(b)$ Note that $\T$ is invertible since $\alpha\in\F_q^\ast$. Then we can conclude by [@TT1 Proposition 25].

Now, we are able to show in an easy and direct way that, similarly to the commutative case, the dual code of a skew $\alpha$-cyclic code is a skew $\alpha^{-1}$-cyclic code. This result was first presented and proven in very different forms in [@B2 Theorem 8], [@B1 Theorem 1] and [@FG Theorem 6.1].

\[GMaruta:1e\] Let $\mathscr{C}\subseteq\F_q^n$ be a linear code and take $\alpha\in\F_q^\ast$. Then $\mathscr{C}$ is a skew $\alpha$-cyclic code if and only if its dual code $\mathscr{C}^\perp$ is a skew $\alpha^{-1}$-cyclic code.

Suppose that $\mathscr{C}$ is a skew $\alpha$-cyclic code invariant under the semi-linear map $\Theta\circ A$, that is, $\mathscr{C}\star (\Theta\circ A)\subseteq \mathscr{C}$. By Lemma \[Tironi\](b), we have $$\mathscr{C}^\perp \star \left(\Theta^{-1}\circ ({}^t \! A)_{\theta^{-1}} \right)= \mathscr{C}^\perp.$$ Since $\Theta\circ M_\theta = M\circ \Theta$ for any matrix $M$, by Lemma \[Tironi\](a) we get $$\begin{aligned}
\mathscr{C}^\perp \star \left(\Theta^{-1}\circ ({}^t \! A)_{\theta^{-1}} \right)= \mathscr{C}^\perp &\Longleftrightarrow \mathscr{C}^\perp \star \left(\Theta^{-1}\circ ({}^t \! A)_{\theta^{-1}} \right)^{-1}= \mathscr{C}^\perp\\
 &\Longleftrightarrow \mathscr{C}^\perp \star \left( (({}^t \! 