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Let $A, B, A-B$ be positive definite matrices. How to show $\mathrm{per} A\ge \mathrm{per} B$? Here $\mathrm{per}$ is the permanent function.

Also, if $A$ is $n\times n$ doubly stochastic matrix which is also positive semidefinite. Let $J$ be the matrix with all entries $1/n$. How to show $A-J$ is also positive semidefinite?

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