This research is a study of inequalities in the Lowner partial order for the special kind of Hermitian matrices, namely, positive definite symmetric matrices. The aim is to show the relationship in the form of inequalities in the Lowner partial order involving square, Kronecker products, and Hadamard products of linear combination of matrices. By applying the monotonically increasing property of trace, determinant and eigenvalues of positive definite symmetric matrices to these inequalities, inequalities involving trace, determinant and eigenvalues of those matrices are obtained. The usual arithmetic-geometric means inequality is a special case of these results.

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