Nonnegative definite hermitian matrices with increasing principal minors

A nonzero nonnegative definite hermitian m by m matrix A has increasing principal minors if the value of each principle minor of A is not less than the value each of its subminors. For $m>1$ we show $A$ has increasing principal minors if and only if $A^{-1}$ exists and its diagonal entries are less or equal to 1.