Let $M: \mathbb{R} \to \mathbb{R}^{2 \times 2}$ be the map defined by

\begin{align*} M(a)_{11}=a,\quad M(a)_{12}=1,\quad M(a)_{21}=1,\quad M(a)_{22}=0. \end{align*}

Let

\begin{align*} \frac{p_n}{q_n}=[a_0;a_1,\dots,a_n] \end{align*}

be the convergents of a finite or infinite simple continued fraction, where $a_0 \in \mathbb{Z}$, $a_i \in \mathbb{N}$ for $i \geq 1$, with the standard initial values $p_{-2}=0$, $p_{-1}=1$, $q_{-2}=1$, and $q_{-1}=0$. Then $M(a_0)M(a_1)\cdots M(a_n)$ is the $2\times 2$ matrix whose first row is $(p_n,p_{n-1})$ and whose second row is $(q_n,q_{n-1})$. Consequently,

\begin{align*} p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}. \end{align*}