Find the values of the parameters for which the matrices commute 
I have two matrices:
  $A = \left( \begin{array}{cc}
1 & a \\
b & 6
\end{array} \right)
%
\ \ \ \ B = \left( \begin{array}{cc}
4 & c \\
d & 2
\end{array} \right)$.
  How can I find the values of the parameters $a, b, c, d$ for which the matrices will commute?

I tried calculate $AB$ and $BA$ and compare the matrices elements, but that is not enough to get the result.
I think this have simple solution, but I have difficulties with it.

We have 
$$AB=\begin{pmatrix}
1&a\\
b&6\\
\end{pmatrix}\begin{pmatrix}
4&c\\
d&2\\
\end{pmatrix}=\begin{pmatrix}
4+ad&c+2a\\
4b+6d&bc+12
\end{pmatrix}$$
and 
$$BA=\begin{pmatrix}
4&c\\
d&2\\
\end{pmatrix}\begin{pmatrix}
1&a\\
b&6\\
\end{pmatrix}=\begin{pmatrix}
4+bc&4a+6c\\
d+2b&ad+12
\end{pmatrix}.$$
Hence 
\begin{eqnarray}
ad&=&bc\\
0&=&2a+5c\\
2b+5d&=&0\\
\end{eqnarray}
Thus $a=\frac{-5}{2}c$ and $b=\frac{-5}{2}d$ and thus $\frac{-5}{2}cd=\frac{-5}{2}dc$ is automatic. In other words for all $c,d$ and $a=\frac{-5}{2}c$ and $b=\frac{-5}{2}d$ the matrices commute. 
Although the equations you get are not linear, a solution to the linear second and third equation automatically gives a solution to the non-linear equation! That makes this easy.
