Determine if the matrix is orthogonal

$\displaystyle \left(\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{ar ray}\right)

$

I don't know how to do it. Any help is appreciated! Thanks!

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- Sep 2nd 2009, 04:57 PMLinnusDetermining orthogonal Matrix
Determine if the matrix is orthogonal

$\displaystyle \left(\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{ar ray}\right)

$

I don't know how to do it. Any help is appreciated! Thanks!

- Sep 2nd 2009, 05:02 PMRandom Variable
The inverse of a orthogonal matrix is its transpose.

- Sep 2nd 2009, 05:06 PMLinnus
I'm not sure what that means or how to figure it out...

But here is the whole question: Consider the unit cube in $\displaystyle \Re^3 $ based at the origin in the (+,+,+) octant. Let $\displaystyle \theta $ be an angle. Determine if the matrix is orthogonal. - Sep 2nd 2009, 05:12 PMRandom Variable
$\displaystyle A = \left(\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{ar ray}\right) $

$\displaystyle A^{T} = \left(\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{ar ray}\right) $

$\displaystyle AA^{T} = A^{T}A = \left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{ar ray}\right) = I$

so A is orthogonal - Sep 2nd 2009, 05:24 PMLinnus
Thanks!

Is that the only way to do it? Is there another way you can check by using dot product or determinants? - Sep 5th 2009, 02:50 PMCNUDrew
Depends. If the determinant of a matrix is zero then it's not invertible thus not possibly orthogonal. Other than that computing $\displaystyle AA^{T}$ should not be hard for small matrices.