Inverse Functions of Fractional Functions and the Shape of the Inverse Matrix of a Quadratic Square Matrix
Theorem
- The inverse function of the fractional function $\displaystyle f(x)=\frac { ax+b }{ cx+d }$ is $$ f^{ -1 }(x)=\frac { dx-b }{ -cx+a } $$
- The inverse matrix of the 2x2 square matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $$ \frac { 1 }{ ad-bc } \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} $$
Explanation
It might just be a coincidence, but finding such coincidences is also the joy of mathematics. Even though matrices have been removed from the curriculum, they can still be very useful.
Proof
$$ \begin{align*} & y=\frac { ax+b }{ cx+d } \\ \implies& (cx+d)y=ax+b \\ \implies& (cy+d)x=ay+b \\ \implies& cxy+dx=ay+b \\ \implies& (cx-a)y=-dx+b \\ \implies& y=\frac { -dx+b }{ cx-a } \\ \implies& y=\frac { dx-b }{ -cx+a } \end{align*} $$
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