TITLE: Can a Matrix with positive entries have a negative eigenvalue?
QUESTION [8 upvotes]: It seems intuitive to me that the answer is no, but I can't prove it.

REPLY [4 votes]: Choose randomly and repeatedly a positive matrix $A$ until $d=\det(A)$ is non-zero.  If $d<0$, then $A$ has at least one $<0$ eigenvalue (why ?). If $d>0$, then swap two columns of $A$.