Orthogonal matrices show that the product is also orthogonal

Question

Show that if A and B are two orthogonal n × n matrices, then so is AB

I know orthogonal is when the transpose of the matrix is equal to it's inverse.

Please help

Answer 1

Very simple, $(AB)(AB)^T=(AB)(B^TA^T)=A(BB^T)A^T=AIA^T=I$, then $AB$ is orthogonal

Answer 2

Another characterization is that $Q$ is an orthogonal matrix iff $\|Qx\|_2=\|x\|_2$ for all $x$.

Then $\|ABx\|_2 = \|Bx\|_2 = \|x\|_2$ for all $x$, hence $AB$ is orthogonal.

Answer 3

$(AB)^T=B^TA^T=B^{-1}A^{-1}=(AB)^{-1}$ and so AB is orthogonal.