In this note we prove that $Tr {MN + PQ} ≥ 0$ when the following two conditions are met: (i) the matrices $M, N, P, Q$ are structured as follows $M = A − B, N = B^{−1} − A^{−1}$, $P = C − D, Q = (B + D)^{−1} − (A + C)^{−1}$ (ii) $A, B$ are positive definite matrices and $C, D$ are positive semidefinite matrices.