Proving Orthogonality of Product of Matrices

[Hypnotoad]

How do you prove that the product of two orthogonal matrices is orthogonal? I know that a matrix can be written in component form as $$A=a_{jk}$$ and that for an orthogonal matrix, the inverse equals the transpose so $$a_{kj}=(a^{-1})_{jk}$$ and matrix multiplication can be expressed as $$AB=\Sigma_ka_{jk}b_{kl}$$. I think that is all I need to be using, but I'm not sure where to go from there.

 

[Galileo]

Think of some other characteristic of orthogonal matrices.
Think about determinants in particular.

 

[shmoe]

the inverse equals the transpose so $$a_{kj}=(a^{-1})_{jk}$$

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

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?

 

[Hypnotoad]

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 $$|AB|=|A||B|$$, 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.

 

[shmoe]

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 $$AB$$ is orthogonal, you can show directly that $$(AB)^{-1}=(AB)^{T}$$. What is $$(AB)^{T}(AB)$$?

 

[Galileo]

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

You're right. I was so totally confused