This question comes from a model in economy developed by Leontief.
Definition If for every non-negative vector $d$, there is a non-negative vector $x$ satisfying $(I-T)x=d$, we say $T$ is feasible.
If for every non-negative vector $v$, there is a non-negative vector $p$ satisfying $(I-T)^{\mathrm{T}}p=v$, we say $T$ is profitable.
Problem Prove that if $T$ is feasible (or profitable), then $\rho(T)<1$.
Here $\rho(T)$ stands for spectual radius, i.e., the maximal norm of eigenvalues of $T$.
I wonder where to get start with. Any help would be appreciated!