non-negative matrix satisfying two conditions

Question

A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$.

I want to find a non-negative matrix $B$ satisfying the following two conditions:

(1) $(I-B)^{-1}$ exists but not non-negative. Here $I$ is the identity matrix.

(2) There is a non-zero and non-negative vector $\vec{d}$ such that $(I-B)^{-1}\vec{d}\ge 0$.

I tried all the $2\times 2$ matrices, but it did not work. I conjecture that such a $B$ does not exist, but don't know how to prove it.

Answer 1

For example $B=diag(1/2,2)$; then $(I-B)^{-1}=diag(2,-1)$ and we can choose $d=[1,0]^T$.

Do you understand why such a $B$ works ?