Generators of $GL_2(\mathbb Q) $

Question

Can anybody help me with following problem, please? I have to show that $$GL_2(\mathbb Q) = \left< \begin{bmatrix} a& 0 \\ 0 & 1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & a \\ \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ \end{bmatrix}: a \in \mathbb Q \setminus \{0\} \right>, $$ but no idea here.

Answer 1

Hint: Any matrix in $\mathbf{GL}_2(\mathbb{Q})$ is equivalent under elementary row operations to the identity matrix. Recall how row reduction is related to elementary matrices and observe that all the proposed generators are indeed elementary matrices.