Derive Equation 2.27.

\(\begin{aligned}
& Q_{11}=\frac{S_{22}}{S_{11} S_{22}-S_{12}^2}=\frac{E_1}{1-v_{12} v_{21}} \\
& Q_{12}=-\frac{S_{12}}{S_{11} S_{22}-S_{12}^2}=\frac{v_{12} E_2}{1-v_{12} v_{21}}=Q_{21} \\
& Q_{22}=\frac{S_{11}}{S_{11} S_{22}-S_{12}^2}=\frac{E_2}{1-v_{12} v_{21}} \\
& Q_{66}=\frac{1}{S_{66}}=G_{12}
\end{aligned} \tag{2.27}\)