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

Question

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.

Answer 1

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.