Math Help - orthogonal matrices

1. orthogonal matrices

Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix.

I think I need to show that the columns of A are orthonormal to each other, but I don't know how.

2. Originally Posted by Jskid
Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix.

I think I need to show that the columns of A are orthonormal to each other, but I don't know how.
You often pick the hardest characterization to use. Try picking another one--solution below--use at your own risk

Spoiler:

You sure you want to look so soon?

Spoiler:

Fine, do it

Spoiler:

Isn't it true that $A$ is real orthogonal if and only if $A^{-1}=A^\top$? So that $(AB)^{-1}=B^{-1}A^{-1}=B^\top A^\top=(AB)^\top$?

3. A "tiny" alternative:

$M\in\mathbb{R}^{n\times n}$ is orthogonal iff $MM^{t}=I$

So, $(AB)(AB)^t=\ldots =I$