Intro Math Matrices problem Find all $2 \times 2$ matrices $A= \begin{pmatrix} a &amp; b \\ c &amp; d \end{pmatrix}$ such that $AB= BA $ for all $2\times2 $ matrices B.

I am stumped as to how to approach this question overall. I am not sure if its asking for me to use some theorem or if some easier solution is present.
Let
$$A=\pmatrix{a&amp;b\cr c&amp;d\cr}$$
and suppose that $AB=BA$ for all $B$.  Making a particular choice for $B$ we have
$$\pmatrix{a&amp;b\cr c&amp;d\cr}\pmatrix{1&amp;0\cr0&amp;0\cr}
  =\pmatrix{1&amp;0\cr0&amp;0\cr}\pmatrix{a&amp;b\cr c&amp;d\cr}\ ,$$
so
$$\pmatrix{a&amp;0\cr c&amp;0\cr}=\pmatrix{a&amp;b\cr0&amp;0\cr}$$
which shows that $b=0$ and $c=0$.  Similarly,
$$\pmatrix{a&amp;0\cr 0&amp;d\cr}\pmatrix{0&amp;1\cr0&amp;0\cr}
  =\pmatrix{0&amp;1\cr0&amp;0\cr}\pmatrix{a&amp;0\cr 0&amp;d\cr}$$
gives $a=d$.  Therefore
$$A=\pmatrix{a&amp;0\cr0&amp;a\cr}=aI\ .$$
We have shown that these are the only possibilities for $A$.  This does not mean they actually work for all $B$, but it is easy to check that in fact they do.
In General, matrix multiplication is not commutative.
AB = BA holds only in case if either A or B is an Identity Matrix, or B is the Inverse of A.