If for the matrix A=pmatrix1&-&pmatrix, AA^T=I_2, then the value of ^4+^4 is

Question

If for the matrix $A=\begin{pmatrix}1&-\alpha\\alpha&\beta\end{pmatrix}$, $AA^T=I_2$, then the value of $\alpha^4+\beta^4$ is

Accepted Answer

$AA^T=\begin{pmatrix}1+\alpha^2 & \alpha-\alpha\beta\ \alpha-\alpha\beta & \alpha^2+\beta^2\end{pmatrix}=I_2$.

From $(1,1)$: $1+\alpha^2=1\Rightarrow\alpha=0$. From $(1,2)$: $\alpha(1-\beta)=0$ (automatically satisfied). From $(2,2)$: $\alpha^2+\beta^2=1\Rightarrow\beta^2=1\Rightarrow\beta=\pm1$.

$\alpha^4+\beta^4=0+1=1$.

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