A math teacher taught a shortcut for calculating the inverse or the derivative of a linear rational function of the form.
$$ R(x) = \frac{ax+b}{cx+d} $$
By first writing it in a matrix form,
$$ A= \begin{bmatrix}a & b\\c & d\end{bmatrix}$$
Then, the inverse of linear function is given by,
$$ adj (A) $$
and the derivative is given by
$$ \frac{det(A)}{(cx+d)^2}$$
Why is that? How did he come up with those formulas? I am having trouble trying to find the reference, or source of this. I know that they are true, but I'd like to know how he this was found, especially, the inverse formula.
I don't know how the teacher came up with those formulas but I can prove them.
First, if $$y=\frac{ax+b}{cx+d}$$ then $$x=\frac{dy-b}{-cy+a}.$$ So, the corresponding matrix is
$$\begin{bmatrix}d&-b\\-c&a\end{bmatrix}.$$ At the same time
$$\operatorname{adj}\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}d&-c\\-b&a\end{bmatrix}^T=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}.$$
Second, the derivative of $y$ is
$$y'=\frac{da-bc}{(cx+d)^2}$$
and the determinant of $A$ is exactly $da-bc$.