Rotation in differnet basis (change of basis)

Question

Let $\mathcal B $= $\left\{\left[ \begin{matrix} 2\\ 3\\ \end{matrix} \begin{matrix} 3\\ 1\\ \end{matrix} \right] \right\}$ be the basis
And i have a rotation Matrix $\mathsf A$ rotate 60 degree: $\left[ \begin{matrix} \cfrac{1}{2} & \cfrac{{\sqrt 3}}{2} \\ -\cfrac{{\sqrt 3}}{2} & \cfrac{1}{2} \\ \end{matrix} \right] $
Vector $\vec x = \begin{bmatrix} 21\\ 14\\ \end{bmatrix}$

since $\hat A=BDB^{-1}$ A hat is a transformation matrix for above basis (rotate 60 degree)



Question 1:
How to find A hat? is it just use $\mathsf A$ as $\mathsf D$ and calculate with formula $\hat A=BDB^{-1}$?

Question 2:
If u don't know the transformation Matrix both D and $\hat A$ only given Basis ,one vector and one image(the result of a vector under transformation it is $\hat A$$\vec x$ in this case )is it possible to find D and $\hat A$?