$A =\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right] \quad AA ^{ T }= I _{2}$

$\Rightarrow\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right]\left[\begin{array}{cc}1 & \alpha \\ -\alpha & \beta\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}1+\alpha^{2} & \alpha-\alpha \beta \\ \alpha-\alpha \beta & \alpha^{2}+\beta^{2}\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow \alpha^{2}=0 \;and\; \beta^{2}=1$

$\therefore \alpha^{4}+\beta^{4}=1$