A sufficient condition for the monotonicity of a positive definite matrix (Q1395200)

The author considers the set of real matrices \(M_n\) of order \(n\geq{2}\) and \(M_n^+\subset{M_n}\) the set of real symmetric positive definite matrices. Monotonicity of a matrix \(A\in{M_n^+}\) (i.e., \(A\) is nonsingular and all entries in \(A^{-1}\) are nonnegative) is proved under the condition that all of its principal submatrices of a fixed order are monotone.