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Problem: Show that the matrix $$A=\begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & -1 \end{bmatrix}$$ represents a rotation $R_\theta$ about a line $l$ through the origin followed by a reflection $H_l$ about the plane through the origin that is perpendicular to the line $l$. You have to find the standard matrices of $R_\theta$ and $H_l$.

My Attempt: I calculated the determinant of A which was -1. I know that A is a orthogonal matrix, and since it has a determinant of -1, it involves reflection. However, I can't think of a way to find the line $l$.
Hints please?

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  • $\begingroup$ Very much similar to a permutation matrix $\endgroup$ – UmbQbify Jun 30 at 11:49
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    $\begingroup$ @ user675453 I'm actually not familiar with the concept. Are you talking about the P matrix in PLU decomposition? Also, if it is though, can you walk me through it? $\endgroup$ – Joshua Woo Jun 30 at 12:48
  • $\begingroup$ yeah. The Matrix containing only one 1 in each row and column. It exactly like that, with opposite vectors (reflection?). I'm not very good at linear algebra. $\endgroup$ – UmbQbify Jun 30 at 13:20
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Hint: $A$ looks a lot like the rotation matrix $$ R_{\theta} = \pmatrix{0&-1&0\\1&0&0\\0&0&1}. $$

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