How $a+bi$ becomes $\left(\matrix{a & -b\\b & a}\right)$? How do you change something into a matrix? E.g) 

How does $2+3i$ become $\left(\matrix{2 & -3\\3 & 2}\right)$

Is  there a specific rule for changing an input into a matrix that has to be learned? Or how is it done?
Thank you.

$$\begin{align}\color{blue}{(a+bi)\cdot}(x+yi)&=(ax-by)+(ay+bx)i\\
\color{blue}{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\cdot}\begin{bmatrix}x\\y\end{bmatrix}&=\begin{bmatrix}ax-by\\ay+bx\end{bmatrix}
\end{align}$$
Do you see the connection? The correspondence between black items is maybe most clear; those are best thought of as elements. The blue items are best thought of as actions (multiplication by ...).
