Proof that the product of a symmetric matrix and its transpose is also symmetric

Question

If A is a symmetric matrix, then verify that A×A' (transpose) and also A'×A is also symmetric

Answer 1

Note that in general $$(AB)'=B'A'$$

For a symmetric matrix we have $A'=A$

$$AA'=A'A=A^2$$ $$(AA')'=A''A=A^2=AA'$$ $$(A'A)'=A'A''=A^2=A'A$$