Proving that the product of orthogonal matrices is also orthogonal.

I'm not sure how to do this. I tried making two general 2 x 2 matricies like so:

To be orthogonal, these equations must be satisfied (via dot product):

ab + cd = 0

ef + gh = 0

Multiplying the two matricies, and then finding the dot product of the result yeilds:

The best I could do was show that:

I think I'm attempting this wrong. :/ Any advice?

Re: Proving that the product of orthogonal matrices is also orthogonal.

is orthogonal iff . So, if are orthogonal then, which implies orthogonal.