If for the matrix,

Question:

If for the matrix, $\mathrm{A}=\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right], \mathrm{AA}^{\mathrm{T}}=\mathrm{I}_{2}$, then

the value of $\alpha^{4}+\beta^{4}$ is :

  1. 4

  2. 2

  3. 3

  4. 1


Correct Option: , 4

Solution:

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

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

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

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

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

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