Let $\displaystyle B = \begin{bmatrix}1&0\\0&0\end{bmatrix}, ~C=\begin{bmatrix}0&1\\0&0\end{bmatrix}$.

Find all 2x2 matrices A such that $\displaystyle AB=BA$ and $\displaystyle AC=CA$.

My analytic solution to this problem was clearly wrong as I got the single zero-matrix as a result. Clearly if A is the identity then those conditions are satisfied, but I don't know the analytic way to solve it correctly.