matrix construction

Question

Given any matrix $A$, can one construct a matrix $B$ such that

1. $B$ is nonnegative and the spectral radius of $B$ is strictly less than 1

2. the determinant of $A$ is equal to the first entry of $B^*$ where $B^*=(I-B)^{-1}$.

Here, by constructing, I certainly do NOT mean "trivial" ones like first computing the determinant of $A$. From the complexity perspective, I want a Logspace algorithm.

Thanks.