1. ## Determining 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!

2. The inverse of a orthogonal matrix is its transpose.

3. 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.

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

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

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

so A is orthogonal

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

6. 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.