Determining orthogonal Matrix

**Linnus** · September 2nd 2009, 04:57 PM

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!

**Random Variable** · September 2nd 2009, 05:02 PM

The inverse of a orthogonal matrix is its transpose.

**Linnus** · September 2nd 2009, 05:06 PM

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.

**Random Variable** · September 2nd 2009, 05:12 PM

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

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

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

so A is orthogonal

**Linnus** · September 2nd 2009, 05:24 PM

Thanks!
Is that the only way to do it? Is there another way you can check by using dot product or determinants?

**CNUDrew** · September 5th 2009, 02:50 PM

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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