Some Properties of M-Matrices

Recently, Markham has proved the following property of M-matrices. Theorem A Suppose A is an M-matrix with the property that for some positive integer p, A~p=L, where L is a lower-triangular matrix. Then A is a lower-triangular matrix. Clearly Theorem A is also true for an upper-triangular matrix. The purpose of this note is to generalize Theorem A to a block upper triangular matrix. If A is a real square matrix with nonpositive off-diagonal elements, each of the following is a necessary and sufficient condition for A to be an M-matrix.