Verify the differentiablity of trace and determinant function

Question

$f: \text{GL}_n(\mathbb R) \times \mathbb M_n(\mathbb R) \to R, f(A,B) = \text{tr}(A^{-1}B)$

$g: \mathbb M_n(\mathbb R) \to \mathbb M_n(\mathbb R), g(A) = \det(A)$

Are $f,g$ differentiable? (Use the definition of differentiation)

We need to find a linear continuous application $L$ such that: $$f(A+H, B+H) = f(A,B) + L_{(A,B)}(H,H) + ||(H,H)|| \epsilon(H,H),$$ where $\epsilon(H,H) \to 0$, as $(H,H) \to (0,0)$

What is the relation between: $\text{tr}((A+H)^{-1}(B+H))$ and $\text{tr}(A^{-1}B)$?

Same question for $\det(A+H)$ and $\det(A)$.

Answer 1

Hint

Use the series expansion for $(I+X)^{-1} = I - X + o(X^2)$

Answer 2

Hint: you can use the fact that determinants are just linear combinations in $\mathbb{R}^{n^2}$. Show for $\mathbb{R}$, then apply induction.

Similarly, trace is just a sum of projections, i.e., a composition of continuous functions.