Sign compatible expressions for minors of the matrix $I-A$

Abstract:

Let $A = ({a_{ij}})$ be an $n \times n$ nonnegative matrix having row sums less than or equal to one. This paper shows that the ijth minor of $I - A$ can be expressed as \[ {( - 1)^{i + j}}\sum {\Pi {r_k}{a_{pq}}} \] where \[ {r_k} = 1 - \sum \limits _{s = 1}^n {{a_{ks}}} \] and each $\Pi {r_k}{a_{pq}}$ is a product of exactly $n - 1$ numbers taken from ${r_k},{a_{pq}}$ for $k,p,q = 1, \ldots ,n$. This theorem is then used to obtain perturbation results concerning the matrix $I - A$.