This question is not homework. As part of a different problem, I had the idea of constructing a matrix where, when computing the determinant, every term in the sum is nonnegative. Specifically, if $A$ is $n$x$n$ and
$$\det(A) = \sum_{\text{permutations } \sigma}\text{sign}(\sigma)\prod_{i=1}^n A_{i,\sigma(i)} $$ then I am asking whether if $A$ is positive definite then $$ \text{sign}(\sigma)\prod_{i=1}^n A_{i, \sigma(i)} \ge 0 $$ for all permutations $\sigma: \{1,\dots,n\} \to \{1,\dots,n\}$.