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For positive integers $u$ and $v$, let $L_u=\begin{bmatrix} 1 & 0 \\ u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}$. Let$S_{u,v}$ be the monoid generated by $L_u$ and $R_v$, and $G_{u,v}$ be thegroup generated by $L_u$ and $R_v$. In this paper we expand on acharacterization of matrices $M=\begin{bmatrix}a & b \\c & d\end{bmatrix}$ in$S_{k,k}$ and $G_{k,k}$ when $k\geq 2$ given by Esbelin and Gutan to $S_{u,v}$when $u,v\geq 2$ and $G_{u,v}$ when $u,v\geq 3$. We give a simple algorithmicway of determining if $M$ is in $G_{u,v}$ using a recursive function and theshort continued fraction representation of $b/d$.

Subgroups of $SL_2(\mathbb{Z})$ characterized by certain continued fraction representations