Green Potassium Manganate and Sodium Hydroxide Equation
$$\ce{ 3K2MnO4 + 2NaOH + 2H2O -&gt; MnO2 +2NaMnO4 + 6KOH }$$

As you said the brown solid A is $\ce{MnO2}$ while the purple solution is due to the permanganate ion .