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[Community Question] Linear-algebra: non-negative matrix satisfying two conditions

One of our user asked:

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.


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