Let us assume that $A\in\mathbb{R}^{n\times n}$. It is known that since $A^TA$ is symmetric positive semi-definite, $\mathrm{det}(A^TA)$ is non-negative. I would like to know what is the importance of $\mathrm{det}(A^TA)$ in practical discussions?
Moreover, can we draw a specific conclusion about the relationship between $\mathrm{det}(A^TA)$ and the entries of A? For example, can we conclude that if the amount of $\mathrm{det}(A^TA)$ tends to zero, then the entries of $A$ also tends to zero?