My group is GL_{2}(R), which by definition is the set of all 2x2 invertible matrices with real entries under matrix multiplication. Therefore, by its definition every element in GL_{2}(R) has an inverse.

Formally, given a matrix A = $\begin{pmatrix}a & b \\c & d \end{pmatrix}$, A^{-1} = $\frac{1}{ad-bc}\begin{pmatrix}d & -b \\-c & a \end{pmatrix}$

Therefore, as long as ad-bc does not equal zero, A will have an inverse.

Therefore, as long as the determinant of A does not equal zero, A will have an inverse.

This emplies my group is the set of all 2x2 matrices with determinant not equal to zero.