Lower bound for the size of a determinant

Question

Given a matrix $A$ in $\mathbb{R}^{n \times n}$ and let $a_{\min} = \min_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}} \{ |A_{ij}| \}$. Does $$ a_{\min} \leq \det(A) $$ always hold?

Answer 1

Hint: Consider the matrix $\begin{pmatrix} \epsilon & 0 \\ 0 & \epsilon \end{pmatrix}$. (This works only if you ignore zero entries, see user1551's comment.)

Another hint (following Famous Blue Raincoat's comment): Consider the matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$.