Let such that then, (I'm assuming you give this space the operator norm):

and we want this to tend to zero as tends to zero, which is clearly satisfied if we define , this is clearly linear, and from which it follows that is a bounded linear operator, and as such is the derivative of at doing this for all we have a map: where .

The second one is a little trickier since the function is only defined in the subset of nonsingular matrices, and you would have to prove first that this subset is open.