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  4. If for the matrix, A = [1&-α α&β], AAT = I2, then the value of α4 + β4 is

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Q. If for the matrix, $A = \begin{bmatrix}1&-\alpha\\ \alpha&\beta\end{bmatrix}, AA^{T} = I_{2}, $ then the value of $\alpha^4 + \beta^4$ is

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Solution:

$A = \begin{bmatrix}1&-\alpha\\ \alpha&\beta\end{bmatrix}, AA^{T} = I_{2} $
$\Rightarrow \begin{bmatrix}1& -\alpha\\ \alpha&\beta\end{bmatrix}\begin{bmatrix}1&\alpha\\ -\alpha&\beta\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix} $
$\Rightarrow \begin{bmatrix}1+\alpha^{2}&\alpha-\alpha\beta\\ \alpha-\alpha\beta&\alpha^{2}+\beta^{2}\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix} $
$\Rightarrow \alpha^2 = 0 $ & $\beta^2 = 1$
$\therefore \alpha^4 + \beta^4 = 1$