In this paper, we investigate a class of matrices, called F-matrices, which contains symmetric positive semidefinite matrices, totally nonnegative matrices, T-matrices and M-matrices. We establish some ineqalities that are improvement of Hadamard's inequality for F-matrices. We prove that det A = a 11… ann if and only if any diagonal line except the main diagonal line of A has at least a zero. And we characterize an F-matrix A satisfying det A = a11… ann by the pattern of zero of A. Our results generalize the known results on Hadamard's Inequality.
Tamkang Journal of Mathematics = 淡江數學 28(1), pp.33-37