Let be the space of real matrices. Show that the function
(where denotes the usual Euclidean norm on ) defines a norm on . Show also that this norm satisfies for all and , and that if then all entries of have absolute value less than . Deduce that any function such that is a polynomial in the entries of is continuously differentiable.
Now let be the mapping sending a matrix to its determinant. By considering as a polynomial in the entries of , show that the derivative is the function . Deduce that, for any is the mapping , where is the adjugate of , i.e. the matrix of its cofactors.
[Hint: consider first the case when is invertible. You may assume the results that the set of invertible matrices is open in and that its closure is the whole of , and the identity .]