Easy method to show orthogonality of a matrix

[dapias09]

Hi, I know how the properties of an orthogonal matrix, the transpose ot the matrix is equal to its inverse. The problem is that the teacher gaves me a 3x3 matrix expressed in terms of many cosines and sines of three angles, I want to know how can I prove that the matrix is orthogonal without having to do the product between the matrix and its transpose.

Thanks for your help.

 

[micromass]

You can always check that the columns form an orthonormal basis. This implies that your matrix is orthogonal. So you only need to calculate a few inner products.

 

[Ben Niehoff]

You can always check that the columns form an orthonormal basis. This implies that your matrix is orthogonal. So you only need to calculate a few inner products.

But this is actually the same as multiplying the matrix with its transpose. In fact it's the exact same sequence of operations.

OP, I don't think there are any shortcuts to this problem unless you know some fancy group theory. Just multiply it out. All the sines and cosines should either cancel or combine into \sin^2 a + \cos^2 a = 1.