Linear Algebra Done Right · Problem 9C.16

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

T \in \mathcal{L}(V)

. Define

g\colon \mathbf{F} \to \mathbf{F}

g(x) = \det(I + xT)

. Show that

g'(0) = \operatorname{tr} T

. Look for a clean solution to this exercise, without using the explicit but complicated formula for the determinant of a matrix.