Norm of a matrix and lower bound for its determinant

Question

Assume that $M$ is a positive constant, $A=[a_{ij}]$ is a matrix, and $\vert a_{ij}\vert \geq M $ for all $1\leq i,j \leq n$. Also, assume that $\det(A) \neq 0$ .Can we conclude that there exists a constant $C > 0$ such that $\det(A) \geq CM$?

Answer 1

No, you cannot. For instance, consider a matrix where every element is equal to $M$; its determinant will be zero.

Answer 2

No.

The determinant is the product of the eigenvalues. You can easily make a matrix with positive entries and negative eigenvalues. For instance, take a $2\times 2$ matrix $$\begin{pmatrix} 1 & 9 \\1 & 1 \end{pmatrix}$$