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Inverse Functions of Fractional Functions and the Shape of the Inverse Matrix of a Quadratic Square Matrix 📂Matrix Algebra

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 f(x)=ax+bcx+d\displaystyle f(x)=\frac { ax+b }{ cx+d } is f−1(x)=dx−b−cx+a f^{ -1 }(x)=\frac { dx-b }{ -cx+a }
  • The inverse matrix of the 2x2 square matrix [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} is 1ad−bc[d−b−ca] \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

y=ax+bcx+d  ⟹  (cx+d)y=ax+b  ⟹  (cy+d)x=ay+b  ⟹  cxy+dx=ay+b  ⟹  (cx−a)y=−dx+b  ⟹  y=−dx+bcx−a  ⟹  y=dx−b−cx+a \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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