How prove or refute $\diamond \Box A$ → A characterizes symmetry

Question

Can some of you nice people help me and show me how to prove $\diamond \Box A$ → A characterizes symmetry.

I really appreciate it Bests

Answer 1

Hints:

1. If $(W, R)$ is a frame with $R$ symmetric, assume $(W, R, p), v\Vdash \diamond\Box A$ and deduce that $A$ is valid at $v$.
2. If $(W,R)$ is not symmetric, there are $v, w\in W$ such that $vRw$ but not $wRv$, then define a valuation $p$ (of $A$) that makes $v\Vdash \diamond\Box A$ but $A$ is false at $v$.