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For the matrix \(\begin{bmatrix}\alpha & \beta & \gamma\\ \beta & \alpha & -\gamma\\ \gamma & -\gamma & \beta\end{bmatrix}\) to be orthogonal, which of the following is true?

1. \(\alpha=\pm\frac{1}{\sqrt2}\)
2. \(\beta=\pm\frac{1}{\sqrt3}\)
3. \(\gamma=\pm\frac{1}{\sqrt2}\)
4. \(\beta=\pm\frac{1}{\sqrt6}\)

Correct answer: \(\beta=\pm\frac{1}{\sqrt3}\)

Solution

For an orthogonal matrix, rows are orthonormal. The dot product of the first and second rows gives \(2\alpha\beta+\gamma^2=0\), and the norm conditions give equations involving \(\alpha,\beta,\gamma\). Solving the system leads to \(\beta^2=\frac13\), so \(\beta=\pm\frac1{\sqrt3}\).

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