Answer
$A=\left[\begin{array}{ccc}\sqrt{3} / 2 & -1 / 2 & 0 \\ -1 / 2 & -\sqrt{3} / 2 & 0 \\ 0 & 0 & 1\end{array}\right]$
Work Step by Step
\[
R_{30^{\circ}}=\frac{1}{2}\left[\begin{array}{cc}
\sqrt{3} & -1 \\
1 & \sqrt{3}
\end{array}\right]
\]
To construct the $2 \mathrm{D}$ matrix which rotates points by $30^{\circ},$ we note that the rotation sends $\mathbf{e}_{1}$ to $\left(\cos \left(30^{\circ}\right), \sin \left(30^{\circ}\right)\right)=(\sqrt{3} / 2,1 / 2)$ and $\mathbf{e}_{2}$ to
\[
\begin{array}{c}
\left(-\sin \left(30^{\circ}\right), \cos \left(30^{\circ}\right)\right)=(-1 / 2, \sqrt{3} / 2) \\
F_{x-\text { axis }}=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
\end{array}
\]
A reflection through the $x$ -axis doesn't move $\mathbf{e}_{1}$ at all (since it's on the $x$ -axis) and sends $\mathbf{e}_{2}$ to $-\mathbf{e}_{2}$.
\[
\begin{aligned}
A_{2 \times 2} &=F_{x \text { -axis }} \cdot R_{30^{\circ}} \\
&=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]\left[\begin{array}{cc}
\sqrt{3} / 2 & -1 / 2 \\
1 / 2 & \sqrt{3} / 2
\end{array}\right] \\
&=\left[\begin{array}{cc}
\sqrt{3} / 2 & -1 / 2 \\
-1 / 2 & -\sqrt{3} / 2
\end{array}\right]
\end{aligned}
\]
To do the rotation first and then the reflection, we compose the matrices with $R_{30^{\circ}}$ on the right (nearest the transformed vector $\mathbf{x})$ and $F_{x \text { -axis }}$ on the left.
$A=\left[\begin{array}{ccc}\sqrt{3} / 2 & -1 / 2 & 0 \\ -1 / 2 & -\sqrt{3} / 2 & 0 \\ 0 & 0 & 1\end{array}\right]$
