If $G = \{ A \in GL(3,\mathbb R): Ax = x \}$, where $x = \begin{bmatrix} 1 \\ 0\\ 0 \\ \end{bmatrix}$, describe all orthogonal matrices in $G$.
2026-03-27 04:16:38.1774584998
Describe all orthogonal matrices in $G$
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An orthogonal matrix represents an isometry of Euclidean space - it preserves the magnitude of any vector and the dot product of any two vectors. $G$ consists of the orthogonal $3 \times 3$ matrices which also leave the $x$ axis unchanged.
Your example is a family of orthogonal matrices, but they do not leave the $x$ axis unchanged. Instead, they map the $z$ acis to the $y$ axis.