Let A be an \(n\times n\) matrix with all powers having all principal minors positive. It is not known if these conditions force the eigenvalues of A to be all positive. The authors prove that conclusion for \(n\leq 4\), and show in general that the eigenvalues of A with the same absolute value form the vertices of a regular polygon of odd order, with one vertex on the positive axis. Actually, the general result is obtained under a weaker assumption on the sums of the principal minors. Some interesting related questions are also discussed.