The first map is just the continuous map , the second is continuous because each coordinate map is continuous, the third is just the continuous multiplication map, the fourth is just the inclusion , and the last just multiplication again.
So, if you know that is a topological group then you're golden because your function is just , which is the composition of two continuous functions. But, is clearly a topological group since the multiplication maps and inversion maps are just rational functions in each coordinate (in fact, this clearly implies that is a Lie group).
Remark: It's not fruitful to fret over whether a given norm is the one that induces your topology. Indeed, is a fin. dim. vector space, and so all norms induce the same topology. But, yes if you fix the usual Euclidean norm for then this norm is carried naturally by the obvious identification to the Frobenius norm.