Let $a_{0}, a_{1}, a_{2}, \ldots$ be positive integers and, for each $n$, let

$$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+} \cdot \ddots+\frac{1}{a_{n}}}$$

with $\left(p_{n}, q_{n}\right)=1$.

Obtain an expression for the matrix $\left(\begin{array}{cc}p_{n} & p_{n-1} \\ q_{n} & q_{n-1}\end{array}\right)$ and use it to show that $p_{n} q_{n-1}-q_{n} p_{n-1}=(-1)^{n+1} .$