Inverse matrix’s eigenvalue?

It’s from the book “linear algebra and its application” by gilbert strang, page 260.

(IA)1=I+A+A2+A3+…

Nonnegative matrix A has the largest eigenvalue λ1<1.

Then, the book says, (IA)1 has the same eigenvector, with eigenvalue 11λ1.

Why? Is there any other formulas between inverse matrix and eigenvalue that I don’t know?

Answer

A matrix A has an eigenvalue λ if and only if A1 has eigenvalue λ1. To see this, note that
Av=λvA1Av=λA1vA1v=1λv

If your matrix A has eigenvalue λ, then IA has eigenvalue 1λ and therefore (IA)1 has eigenvalue 11λ.

Attribution
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