Solution
= Solution
The transpose reverses the product order, so the definition is
$$
C=A(AB)^T=AB^TA^T.
$$
If $A$ and $B$ are <orthogonal matrices>, then $A^TA=AA^T=I$ and $B^TB=BB^T=I$. Hence
$$
C^TC=(ABA^T)(AB^TA^T)
=AB(A^TA)B^TA^T=ABB^TA^T=I.
$$
Thus \b[$C$ is orthogonal]. The original PDF includes the transpose of the entire product $AB$, which is essential to the expression used here.