Why is $(x'x)^{-1}x' = x(x'x)^{-1}$

If $(AB)'=B'A'$ then $(x'x)^{-1}x'$ should be equal to $x((x'x)^{-1})'$ . However most econometrics textbooks say that this is equal to $x(x'x)^{-1}$ . What happened to the transpose of $(x'x)^{-1}$? According to the rule it should have a transpose.

• What does ' mean here? – Cameron Williams Feb 2 '14 at 2:29
• @CameronWilliams: I think ' means transpose – voldemort Feb 2 '14 at 2:31
• Yes, it means transpose. – Sophie Feb 2 '14 at 2:38
• What is the source of this question, is it from a book? Since you're new, my advice is maybe you can clarify your question to avoid getting down voted so much. – grayQuant Feb 2 '14 at 6:15

Let $a=x'x$ and $b=a^{-1}$. Then $a'=(x'x)'=x'x''=x'x=a$ hence $b'=b$. Thus your suggestion to use the transpose of $(x'x)^{-1}$ and the practice of using $(x'x)^{-1}$ instead, are in fact equivalent.