Relations between matrix norm and determinant I was wondering whether there is a way to obtain the determinant of a matrix out of its norm (when the matrix is regular otherwise it is not true). If $A$ is a square matrix of dimension $n\geq 1$, and $\det A\neq 0$, do we have something like?
$$|Ax|_2 \leq \|A\|_{\text{op}} |x|_2 \leq C_n |\det A| \ |x|_2 \leq \cdots$$
or similar? Or maybe a similar estimate for different norms for $A$ and $x$ if needed since many matrix norms are related to eigen- or singular values which are related to the determinant.
Thanks a lot! :)

I don't see any way to obtain an upper bound of $\|Ax\|$ using the determinant, because the norm of a matrix can remain unchanged when the determinant approaches zero. E.g. when $x=(1,0,\ldots,0)$ and $A=\operatorname{diag}(1,t,\ldots,t)$, we have $\|Ax\|_2=1$ but $\det A\to0$ as $t\to0$.
