# Mobius composition and matrices

- learn how to multiply \(\normalsize 2\times 2\) matrices
- see the connection between matrix multiplication and composition of Mobius functions.

## Mobius transformations and matrices

We have seen that a Mobius transformation, or fractional linear transformation, is a function of the form \[\Large{y=f(x)=\frac{ax+b}{cx+d}}\]where the numbers \(\normalsize{a}\), \(\normalsize{b}\), \(\normalsize{c}\) and \(\normalsize{d}\) can be arbitrary (subject to the common sense requirement that the fraction does not end up with a zero in the denominator!) So the function can be encoded by the matrix \[\Large{M = \begin{pmatrix} a & b\\ c & d \end{pmatrix}}.\]This kind of mathematical object is very important in linear algebra.In this somewhat advanced step, you’ll see how the composition of fractional linear transformations connects directly with matrix multiplication.\[\Large{f(x)=\frac{x}{3x+2}}?\]Q1(E): What is the corresponding matrix for the Mobius transformation

## How to multiply matrices

\[\Large{A=\begin{pmatrix} 1 & 2\\ -1 & 5 \end{pmatrix}} \quad \text{and} \quad \Large{B=\begin{pmatrix} 3 & 1\\ 6 & 4 \end{pmatrix}}\]Q2(M): For the matricesfind the products \(\normalsize{AB}\) and \(\normalsize{BA.}\) Note that matrix multiplication is not in general commutative!

## Composition of Mobius transformations

\[\Large{k(x)=\frac{x+1}{3x-2}}\]Q3(M): Find the composition \(\normalsize{k \circ l}\) of the Mobius transformationsand\[\Large{l(x)=\frac{2x-1}{x}}.\]

## Answers

\[\Large{f(x)=\frac{x}{3x+2} \quad\text{is}\quad \begin{pmatrix} 1 & 0\\ 3 & 2 \end{pmatrix} }.\]A1.The matrix for the Mobius transformation\[\Large AB=\begin{pmatrix} 1 & 2\\ -1 & 5 \end{pmatrix}\begin{pmatrix} 3 & 1\\ 6 & 4 \end{pmatrix}= \begin{pmatrix} 1\times 3+ 2\times 6 & 1\times 1+2\times 4\\ -1\times 3+5\times 6 & -1\times 1+5\times 4 \end{pmatrix}=\begin{pmatrix} 15 & 9\\ 27 & 19 \end{pmatrix}.\]A2.One product isThe other product is\[\Large BA= \begin{pmatrix} 3 & 1\\ 6 & 4 \end{pmatrix}\begin{pmatrix} 1 & 2\\ -1 & 5 \end{pmatrix}= \begin{pmatrix} 3\times 1+ 1\times (-1) & 3\times 2+1\times 5\\ 6\times 1+4\times (-1) & 6\times 2+4\times 5 \end{pmatrix}=\begin{pmatrix} 2 & 11\\ 2 & 32 \end{pmatrix}.\]So, \(\normalsize{AB\neq BA}\)!\[\Large{k(x)=\frac{x+1}{3x-2} \quad \text{is} \quad \begin{pmatrix} 1 & 1\\ 3 & -2 \end{pmatrix} }.\]A3.You could find this directly, but we’ll use matrix multiplication to solve this question. The matrix for the Mobius transformationSimilarly the matrix for the Mobius transformation\[\Large{l(x)=\frac{2x-1}{x} \quad \text{is}\quad \begin{pmatrix} 2 & -1\\ 1 & 0 \end{pmatrix} }.\]So, the matrix corresponding to the Mobius transformation\[\Large{ (k \circ l)(x) \quad \text{is}\quad \begin{pmatrix} 1 & 1\\ 3 & -2 \end{pmatrix}\begin{pmatrix} 2 & -1\\ 1 & 0 \end{pmatrix}=\begin{pmatrix} 3 & -1\\ 4 & -3 \end{pmatrix}}.\]Therefore, the composition is\[\Large{(k\circ l)(x) = \frac{3x-1}{4x-3}}.\]

#### Maths for Humans: Linear, Quadratic & Inverse Relations

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