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- Thread starter Hypnotoad
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Galileo

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Think of some other characteristic of orthogonal matrices.

Think about determinants in particular.

Think about determinants in particular.

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shmoe

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Hypnotoad said:the inverse equals the transpose so [tex]a_{kj}=(a^{-1})_{jk}[/tex]

As you've written it, this is incorrect. You don't take the inverse of the entries. If [tex]A=[a_{jk}][/tex] is orthogonal then [tex]A^{-1}=A^{T}=[a_{kj}][/tex].

There's no need to go into the entries though. You can directly use the definition of an orthogonal matrix. Answer this question: what do you have to do to show (AB) is orthogonal?

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Galileo said:Think of some other characteristic of orthogonal matrices.

Think about determinants in particular.

Well the determinant of an orthogonal matrix is +/-1, but does a determinant of +/-1 imply that the matrix is orthogonal? I know that the determinant is distributive [tex]|AB|=|A||B|[/tex], so the determinant of the product does have to be +/-1, but I don't know if that is sufficient to show that a matrix is orthogonal.

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shmoe

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Hypnotoad said:Well the determinant of an orthogonal matrix is +/-1, but does a determinant of +/-1 imply that the matrix is orthogonal?

No, it doesn't. There are matrices with determinant +/- 1 that are not orthogonal.

To show [tex]AB[/tex] is orthogonal, you can show directly that [tex](AB)^{-1}=(AB)^{T}[/tex]. What is [tex](AB)^{T}(AB)[/tex]?

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Galileo

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You're right. I was so totally confusedshmoe said:No, it doesn't. There are matrices with determinant +/- 1 that are not orthogonal.

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