[$\displaystyle \star \star $] Let $\displaystyle A,B$ be any $\displaystyle 2 \times 2$ matrices with entries from $\displaystyle \mathbb{C}$ and $\displaystyle AB=BA.$ Let $\displaystyle t_A$ and $\displaystyle I$ denote the trace of $\displaystyle A$ and the $\displaystyle 2 \times 2$ identity matrix respectively.

Find some $\displaystyle \alpha, \beta \in \mathbb{C}$ such that $\displaystyle (2A - t_AI)B=\alpha A + \beta I.$