If for the matrix, $\mathrm{A}=\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right], \mathrm{AA}^{\top}=\mathrm{I}_{2}$, then the value of $\alpha^{4}+\beta^{4}$ is:
Correct Option: 1
$\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right]\left[\begin{array}{cc}1 & \alpha \\ -\alpha & \beta\end{array}\right]=\left[\begin{array}{cc}1+\alpha^{2} & \alpha-\alpha \beta \\ \alpha-\alpha \beta & \alpha^{2}+\beta^{2}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$1+\alpha^{2}=1$
$\alpha^{2}=0$
$\alpha^{2}+\beta^{2}=1$
$\beta^{2}=1$
$\alpha^{4}=0$
$\beta^{4}=1$
$\alpha^{4}+\beta^{4}=1$
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