Kunze Linear Algebra · Problem 6.2.8

Elementary Canonical Forms

Mathematics — Methods · invertibility-identity-minus-product

Problem

Let

A

and

B

n \times n

matrices over the field

F

. Prove that if

(I - AB)

is invertible, then

I - BA

is invertible and

(I - BA)^{-1} = I + B(I - AB)^{-1}A.