I'm tasked with showing that matrix $A$ commutes with every $2\times2$ matrix if and only if $A = \begin{bmatrix}a & 0\\0 & a\end{bmatrix}$ for some a.
I was able to prove in the first direction, assuming that $A = \begin{bmatrix}a & 0\\0 & a\end{bmatrix}$, by just multiplying the matrix by $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ in AB = BA format, and showing how AB does equal BA.
$ (=>)$
$\begin{bmatrix}a & 0\\0 & a\end{bmatrix}\begin{bmatrix}a & b\\c & d\end{bmatrix} = \begin{bmatrix}a & b\\c & d\end{bmatrix}\begin{bmatrix}a & 0\\0 & a\end{bmatrix}$
$\begin{bmatrix}a^2 & ab\\ac & ad\end{bmatrix} = \begin{bmatrix}a^2 & ab\\ac & ad\end{bmatrix}$
But now I have to prove in the other direction, assuming that A already does commute with every $2\times 2$ matrix, and I'm not particularly sure how I can do that. Somehow, I have to show that if A commutes with $\begin{bmatrix}a & b\\c & d\end{bmatrix}$, then A will be $\begin{bmatrix}a & 0\\0 & a\end{bmatrix}$, and I don't know how I can carry that out.
Where can I start? Ideas are appreciated.