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Math Help - Determining orthogonal Matrix

  1. #1
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    Determining orthogonal Matrix

    Determine if the matrix is orthogonal

    \left(\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{ar  ray}\right)<br />

    I don't know how to do it. Any help is appreciated! Thanks!
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  2. #2
    Super Member Random Variable's Avatar
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    The inverse of a orthogonal matrix is its transpose.
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  3. #3
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    I'm not sure what that means or how to figure it out...

    But here is the whole question: Consider the unit cube in  \Re^3  based at the origin in the (+,+,+) octant. Let  \theta be an angle. Determine if the matrix is orthogonal.
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  4. #4
    Super Member Random Variable's Avatar
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     A = \left(\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{ar  ray}\right)

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

     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
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  5. #5
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    Thanks!
    Is that the only way to do it? Is there another way you can check by using dot product or determinants?
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  6. #6
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    Depends. If the determinant of a matrix is zero then it's not invertible thus not possibly orthogonal. Other than that computing  AA^{T} should not be hard for small matrices.
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