# Why can inverse and derivative of linear rational functions be represented in the form of adjoint and determinant respectively?

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.

• just apply quotient rule – JJR May 29 '17 at 15:34
• Well, yeah I know, but what about the inverse? – user3855929 May 29 '17 at 15:40

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 $da-bc$.