A $Z$-matrix is a matrix $A\in\mathbb{R}^{n\times n}$ that can be written in the form $A=sI-B$ where $I$ denotes the identity matrix and $B\geq0$, that is every entry of $B$ is nonnegative. Moreover $A$ is called a $M$-matrix if $A=sI-B$ is a $Z$-matrix and $s\geq\rho(B)$. Clearly by the Perron-Frobenius-Theorem such an $A$ is nonsingular iff $s>\rho(B)$.
In Berman and Plemmons book "Nonnegative Matrices in the Mathematical Sciences" they list a lot of equivalent characterisations of nonsingular $M$-matrices und the condition that the matrix is a $Z$-matrix. I am particularly interested in the following: \begin{align} \text{$A$ non-singular $M$-matrix}\Leftrightarrow \text{$A$ is a $Z$ matrix and every eigenvalue of $A$ has positive real part} \end{align} I am looking for an "elementary" proof that doesn't use the characterisations mentioned above. Clearly, the implication from left to right is trivial, but I couldn't come up with the other direction so far. Does someone have an idea?
The equivalence is trivial because of the fact that if the spectrum of $B$ is given by $\{\lambda_1,...,\lambda_k\}$ then the spectrum of $A=sI-B$ is given by $\{s-\lambda_1,...,s-\lambda_k\}$.