Answer
$\left[\begin{array}{ccc}\frac{1}{2} & -\frac{\sqrt{3}}{2} & 3+4 \sqrt{3} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} & -3 \sqrt{3}+4 \\ 0 & 0 & 1\end{array}\right]$
Work Step by Step
\[
\begin{array}{l}
{\left[\begin{array}{l}
x \\
y \\
1
\end{array}\right] \rightarrow\left[\begin{array}{cccc}
1 & 0 & -6 \\
0 & 1 & -8 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
1
\end{array}\right]} \\
{\left[\begin{array}{l}
x \\
y \\
1
\end{array}\right] \rightarrow\left[\begin{array}{cccc}
\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\
\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{ccc}
1 & 0 & -6 \\
0 & 1 & -8 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{c}
x \\
y \\
1
\end{array}\right]} \\
{\left[\begin{array}{l}
x \\
y \\
1
\end{array}\right] \rightarrow\left[\begin{array}{lll}
1 & 0 & 6 \\
0 & 1 & 8 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{ccc}
\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\
\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{ccc}
1 & 0 & -6 \\
0 & 1 & -8 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
1
\end{array}\right]}
\end{array}
\]
rotation of points accomplished by first translating the figure by $-P$, then rotate about the origin, then again translate back by P.
\[
\left[\begin{array}{ccc}
1 & 0 & 6 \\
0 & 1 & 8 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{ccc}
\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\
\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{ccc}
1 & 0 & -6 \\
0 & 1 & -8 \\
0 & 0 & 1
\end{array}\right]=\left[\begin{array}{ccc}
\frac{1}{2} & -\frac{\sqrt{3}}{2} & 3+4 \sqrt{3} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} & -3 \sqrt{3}+4 \\
0 & 0 & 1
\end{array}\right]
\]
Required matrix for composite transformation.
