Find all matrices $A$ (if any) which commute with $C$ (Just need my work checked)

Question

$A=\begin{bmatrix} a & b\\ c & d\\ \end{bmatrix},\; C= \begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix}$

Here's my attempt:

$AC=CA,$

$AC= \begin{bmatrix} b & 0\\ d & 0\\ \end{bmatrix},\; CA= \begin{bmatrix} 0 & 0\\ a & b\\ \end{bmatrix}$

Therefore, $b=0, d=a$.

Did I approach this correctly? I know it probably seems fairly easy.

Answer 1

That's a correct method. If $b=0, d=a$ then they commute (as you've shown), and if they commute then these conditions hold. So this condition is necessary and sufficient.