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.

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


Second, the derivative of $y$ is


and the determinant of $A$ is exactly $da-bc$.


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