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?

  • $\begingroup$ Very much similar to a permutation matrix $\endgroup$ – UmbQbify Jun 30 at 11:49
  • 1
    $\begingroup$ @ 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

Hint: $A$ looks a lot like the rotation matrix $$ R_{\theta} = \pmatrix{0&-1&0\\1&0&0\\0&0&1}. $$

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.