<s>[INST] All matrices which commute with all $2\times 2$ matrices I would like to find all matrices which commute with all $2\times2$ matrices. 
I started solving problem in this way:  
1) I have this matrix $A$ with real numbers:
$$A=\left[\begin{array}{cc}a &b\\c &d\end{array}\right]$$
2) Matrix which commute with matrix $A$ is matrix $B$:
$$B = \left[\begin{array}{cc}e& f\\ g &h\end{array}\right]$$
3) When i solve the equation $AB=BA$, I get this:
$$\left[\begin{array}{cc}ae+bg&af+bh\\ce+dg&cf+dh\end{array}\right]=\left[\begin{array}{cc}ea+cf&eb+df\\ga+ch&gb+dh\end{array}\right]$$
How I can get the general look of wanted matrix?
 [/INST] we can try four different scenarios with matrices that have 3 zero entires. We can assumed $\mathbf{A}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. We can then try $\mathbf{B}=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$.
Solving for $a,b,c$ & $d$ we get $\mathbf{AB}=\mathbf{BA}$ when $c=0$ and $b=0$.
We can then solve for $\mathbf{B}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ which gives us $\mathbf{AB}=\mathbf{BA}$ when $b=0$ and $c=0$.
For $\mathbf{B}=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ we get $\mathbf{AB}=\mathbf{BA}$ when $c=0$ and when $a=d$.
The last equation, $\mathbf{B}=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, give us $\mathbf{AB}=\mathbf{BA}$ when $b=0$, $c=0$, $a=d$.
These equations imply that $a=d$ and $b=c=0$. this means that $\mathbf{A}=\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$ is the form for all possible solutions for $A$ in which every matrix of the form found commute with all matrices $\bold{B}$.
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