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If $p/q<r/s$ are two rational numbers the $rq-ps>0$ and so if the determinant is $\pm 1$, then in fact $rq-ps=1$.
If $\begin{pmatrix} r & p \\ s & q\end{pmatrix}$ is a matrix with positive integer entries such that $rq-ps=1$, then we can use the Euclid division algorithm to show that it can be uniquely written in the form $A^{a_1}B^{b_1}A^{a_2}B^{b_2}\dots A^{a_k}B^{b_k}$ where $a_i$ and $b_i$ are non-negative integers and all except possibly $a_1$ and $b_k$ are non-zero as well. This is what gives the connection of such matrices with continued fractions. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/210.212.36.65|210.212.36.65]] ([[User talk:210.212.36.65|talk]]) 09:00, 30 March 2012 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
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