The question asks about positive definite matrices, not symmetric matrices. However, if $A^{1/2}$ is a symmetric square root, $A^T = (A^{1/2}A^{1/2})^T=A$, so $A$ is also symmetric. The given solution only works if $A$ and $B$ are symmetric. Unfortunately Federico Poloni's answer doesn't answer the question in general as I thought. If the question intended to ask about symmetric matrices, maybe it should be edited.