Two sequences, So weird !

Algebra Level 5

Two real sequences \( \displaystyle {\{a_{n}\}}_{n=0}^{\infty}\),\( \displaystyle {\{b_{n}\}}_{n=0}^{\infty}\) are defined as follows \( \displaystyle a_{1}=\alpha\) and \( \displaystyle b_{1}=\beta\) ,\[ \displaystyle a_{n+1}=\alpha a_{n}-\beta b_{n}\] \[b_{n+1}=\beta a_{n}+\alpha b_{n}\] For all \(n \ge 2\) . How many non-zero pairs \((\alpha,\beta)\) of real numbers are there such that \( \displaystyle a_{1997}=b_{1}\) and \( \displaystyle b_{1997}=a_{1}\)?

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