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:

$\displaystyle \[

A =

\begin{array}{cc}

a & b \\

c & d \\

\end{array}

\]$

$\displaystyle \[

B =

\left( {\begin{array}{cc}

e & f \\

g & h \\

\end{array} } \right)

\]$

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:

$\displaystyle a^2ef + b^2gh = 0$

$\displaystyle c^2ef + d^2gh = 0$

The best I could do was show that:

$\displaystyle (a^2 - c^2)(ef) = (b^2 - d^2)(ef)$

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

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

$\displaystyle M\in\mathbb{R}^{2\times 2}$ is orthogonal iff $\displaystyle M^tM=I$ . So, if $\displaystyle A,B\in\mathbb{R}^{2\times 2}$ are orthogonal then, $\displaystyle (AB)^t(AB)=\ldots=I$ which implies $\displaystyle AB$ orthogonal.