Let $A$ be an $n\times n$ matrix, does the following equation
$|det(A)|\leq \Pi_{i, j} (|a_{i,j}|+1)$
hold? In particular, when does the equality hold? Can we tighten the inequality?
(I believe this is correct, but want to see a direct proof from the definition of determinant.)
Partial answer: $$|\det(A)| = \left|\sum_{\sigma \in S_n} sgn(\sigma) \prod_{i=1}^n a_{i,\sigma_i}\right|\leq \sum_{\sigma \in S_n}\prod_{i=1}^n |a_{i,\sigma_i}|$$ This last term is the sum of all products of $n$ different entries $|a_{ij}|$, and as the product $$\prod_{i,j}(1+|a_{ij}|)$$ does in particular contain all such terms as summands (pick the $|a_{ij}|$ which appear in one of the above summands, and the in the remaining terms pick $1$ ), you are done.