Show that $GL(2, \Bbb Q)$ is isomorphic to a subgroup of $GL(3, \Bbb Q)$

Question

What should be my approach for this particular question and general approach to prove isomorphism of one group to another ?

Answer 1

Just consider $\mathcal i: GL(2,\Bbb Q) \hookrightarrow GL(3,\Bbb Q)$ given by

$\begin{pmatrix} a_{11} &a_{12} \\ a_{21} &a_{22}\end{pmatrix} \mapsto \begin{pmatrix} 1 &0 &0\\ 0 &a_{11} &a_{12} \\ 0 &a_{21} &a_{22}\end{pmatrix}$