Commutativity of Positive Definite Matrices

Question

How do you show that two matrices, $A$ and $B$ that are positive definite commute?

I know this property is true since the set of positive matrices of size n, $Pos(n,\mathbb{R})$ is a subgroup of general linear group $GL(n,\mathbb{R})$, and hence $(AB)^T=B^TA^T=BA$ must equal $AB$ for the group to be closed under operation of multiplication.

Answer 1

This is not true. Consider a counterexample. Both $$ A=\begin{pmatrix} 1 & 0 \cr 0 & 2 \end{pmatrix},\quad B=\begin{pmatrix} 1 & \frac{1}{2} \cr \frac{1}{2} & 1 \end{pmatrix}, $$ are positive definite, but $AB\neq BA$.