Linear Algebra Done Right · Problem 9C.12

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

S, T \in \mathcal{L}(V)

and

S

is invertible. Define

p\colon \mathbf{F} \to \mathbf{F}

p(z) = \det(zS - T).

Prove that

p

is a polynomial of degree

\dim V

and that the coefficient of

z^{\dim V}

in this polynomial is

\det S