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 $\sin^2 a + \cos^2 a = 1$.