If A and B are orthogonal matrices of the same order, then prove that AB is also an orthogonal Matrix.

Since A and B re orthogonal matrices.  Therefore, 

$A{A}^T=A^{T}A=I$ and $B{B}^T=B^{T}B=I$

now,$\left(AB\right){\left(AB\right)}^T=\left(AB\right)\left(B^T{A}^T\right)$

                                   $=A\left(B{B}^T\right)A^T=AI{A}^T$=$A{A}^T=I$

Similarly, we can prove that

 

$\left(AB{)}^T\right(AB)=I$

$\left(AB\right)(AB{)}^T=I=(AB{)}^T\left(AB\right)$

 

Hence, AB is an orthogonal matrix.