Order of matrix multiplication

Question

My textbook states that $$A^T = (SDS^{T})^{T} = (S^T)^TD^TS^T$$

where S is an orthogonal matrix and D is a diagonal matrix. But shouldn't it be

$$A^T = (SDS^{T})^{T} = S^TD^T(S^T)^T$$? Why and when are we allowed to rearrange the order of the matrices?

Answer 1

It's not rearrangement ; this relation is valid:

$(AB)^T=B^TA^T$

for any two matrices $A,B$.