Directional derivative of the determinant

Question

Please help me find the mistake in my derivation:

Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. Computing the directional derivative we get:

$$\nabla_If(A)=lim_{x\to0}\frac{det(A+Ix)-det(A)}{x}=lim_{x\to0}\frac{-p_A(-x)-(-p_A(0))}{x}=p_A'(0)$$

According to notes of a course i'm taking: $\nabla_If(A)=tr(A)$. Where is my mistake?

EDIT: I just looked again and i got it backwards! It says that the derivative of the determinant at $I$ along any matrix is the trace of the matrix. $$Df_I(A)=lim_{x\to0}\frac{det(I+Ax)-1}{x}=det(A)lim_{x\to0}\frac{det(A^{-1}+xI)-det(A^{-1})}{x}=det(A)p'_{A^{-1}}(0)$$

The last expression is the trace since $p_{A^{-1}}(x)=\frac{(-x)^n}{det(A)}p_A(\frac{1}{x})$

I would delete if I could...

Answer 1

Ok, just try it with a diagonal matrix: $$ A+Ix = diag(a_{11}+x,\ldots, a_{nn}+x)$$ then $$ \det(A+Ix)=\prod_{i=1}^n (a_{ii}+x)$$ Then the derivative is for example if $n=2$ $$ a_{22}+a_{11}$$

Then you can generalize for other matrices.