Suppose is an matrix with real entries such that the diagonal entries are all positive, off diagonal entries are all negative, and the row sums are all positive. Prove that .
Suppose to the contrary that . Then there must exist a nonzero solution to the matrix equation . Let be such a solution, and choose such that is maximized. Using the triangle inequality, we have . Recall that a consequence of the triangle inequality is that for all and . Here, we have . On the other hand, . Thus , a contradiction since .