Yes. Conceptually, all the matrices in the group have the same structure, except for the variable component . So, each matrix is identified by its top-right coefficient, since the other three entries remain constant.
However, let's prove in a more formal way that
is an isomorphism.
First of all, it is injective: suppose . Then, you trivially have , because they are two different matrices:
Secondly, it is trivially surjective: the matrix
is clearly the image of the real number x.
Finally, and its inverse are both homomorphisms: if we consider the usual product between matrices to be the operation for the group G and the real numbers to be an additive group, we have