Question:

Matrices P and Q are members of a set R which is defined as follows:

$\displaystyle R= R= \left \{ \begin{pmatrix} a &b \\ c &d \end{pmatrix} a, b, c, d \epsilon \mathbb{R}, ad-bc=1\right \} $

Solution

$\displaystyle P=\begin{pmatrix} l &k \\ k & l \end{pmatrix} $

$\displaystyle Q = \begin{pmatrix} u & v \\ u & -v \end{pmatrix} $

$\displaystyle QP=\begin{pmatrix} (lu+kv) & (lv-ku) \\ (ku+lv)&(kv-lu)\end{pmatrix}$

since $\displaystyle ad-bc=1$

$\displaystyle ({kv+lu)(kv-lu)}{(ku+lv)(lv-ku)=1} $

$\displaystyle (kv^2-lu^2)-(lv^2-ku^2)=1 $

$\displaystyle (v^2+u^2)(k-l)=1 $

$\displaystyle (v^2+u^2)=k-l $

this is the part i am stuck at

please help me