If for the matrix,
Question:

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:

 

  1. (1) 1

  2. (2) 3

  3. (3) 2

  4. (4) 4


Correct Option: 1

Solution:

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

$1+\alpha^{2}=1$

$\alpha^{2}=0$

$\alpha^{2}+\beta^{2}=1$

$\beta^{2}=1$

$\alpha^{4}=0$

$\beta^{4}=1$

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

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