Determinantal inequalities of positive definite matrices (Q2794811)

The main result of this paper is the inequality NEWLINE\[NEWLINE\det (\sum\nolimits_{i = 1}^m {A_i^{ - 1}} ) \geqslant \det {(\sum\nolimits_{i = 1}^m {{{(A_i^{(1)})}^{ - 1}})\dots\det (} \sum\nolimits_{i = 1}^m {(A_i^{(k)}} )^{ - 1}}),NEWLINE\]NEWLINE where \({A_i}\), \(i = \overline {1,m}\), are positive definite matrices having diagonal blocks \(A_i^{(j)}\), \(1 \leqslant j \leqslant k\), with \(A_i^{(j)}\), \(i = \overline {1,m} \), of the same size for each \(j\). Some other determinantal inequalities related to positive definite matrices are also derived.