About relation in linear polynomial over linear polynomial with matrices.

Question

I was analyzing rational expressions involving linear polynomial over linear polynomial of the form: $${ax+b \over cx+d},\;\text{where $a,b,c,d\in\Bbb R$};$$ Amazingly, these polynomials have some properties related to that of matrices of the form: $$\begin{pmatrix}a&b\\c&d\end{pmatrix};$$ If we consider, $f(x) = \cfrac{ax+b}{cx+d};$ equivalent to $F=\begin{pmatrix}a&b\\c&d\end{pmatrix};$ for instance.

We will find that $f^{-1}(x)$ is equivalent to $\operatorname{adj} F$.

Here's how. $$\begin{align} f(x)=y&=\cfrac{ax+b}{cx+d};&F=\begin{pmatrix}a&b\\c&d\end{pmatrix};\\ \\ (cx+d)y&=ax+b;\\ cxy+dy&=ax+b;\\ x(cy-a)&=-dy+b;\\ x&=\cfrac{-dy+b}{cy-a}&=\cfrac{dy-b}{-cy+a}; \\ \\ f^{-1}(x)&=\cfrac{dy-b}{-cy+a};&\operatorname{adj}F=\begin{pmatrix}d&-b\\-c&a\end{pmatrix}; \end{align}$$ Amazingly, function compositions also are related to matrices. Take a look at this. $$\text{let}\; f(x)=\cfrac{ax+b}{cx+d} \;\text{ and }\; g(x)=\cfrac{px+q}{rx+s};\\ F=\begin{pmatrix}a&b\\c&d\end{pmatrix} \;\text{ and }\; G=\begin{pmatrix}p&q\\r&s\end{pmatrix};\\ \begin{align} f\circ g(x)&=\cfrac{a\cdot g(x)+b}{c\cdot g(x)+d};\\ &=\cfrac{a{px+q\over rx+s}+b}{c{px+q\over rx+s}+d} \times {rx+s\over rx+s};\\ &=\cfrac{a(px+q)+b(rx+s)}{c(px+q)+d(rx+s)};\\ f\circ g(x)&=\cfrac{(ap+br)x+(aq+bs)}{(cp+dr)x+(cq+ds)};\\ \\ FG&=\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}p&q\\r&s\end{pmatrix};\\ FG&=\begin{pmatrix}ap+br&aq+bs\\cq+dr&cq+ds\end{pmatrix}; \end{align}$$

Clearly, if we relate functions $f(x)$ and $g(x)$ with matrices $F$ and $G$. Then, $f^{-1}(x)$ can be related to $\operatorname{adj} F$ and $f \circ g \; (x)$ can be related to $FG$.

Finally, after with all this long story, the main questions are:

1. Was it known before, if yes does it have a name?
2. Why does it happen, does it explain why matrices operations were defined like this? i.e. When creators of Matrix were defining Matrix multiplications and Adjoint Matrix, they were thinking about all this?
3. Main question: Can this be somehow related to 3 quadratic equations somehow using something?

Answer 1

Good noticing! This connection is well known; see the Wikipedia article Möbius transformations.