It's from the book "linear algebra and its application" by gilbert strang, page 260.
$(I-A)^{-1}$=$I+A+A^{2}+A^{3}$+...
Nonnegative matrix A has the largest eigenvalue $\lambda_{1}$<1.
Then, the book says, $(I-A)^{-1}$ has the same eigenvector, with eigenvalue $\frac{1}{1-\lambda_{1}}$.
Why? Is there any other formulas between inverse matrix and eigenvalue that I don't know?