[Math] Is there relationship between magnitude of matrix-vector multiplication and determinant of that matrix

Question

If I have a matrix $A$ and vector $x$ is there such a relationship or something similar involving determinants?
$$\|Ax\| \leq |\det A|\|x\|$$
where the absolute values indicate the usual Euclidean norm?

Answer 1

AGTortorella showed that it cannot be true when $det(A) = 0$, but for $det(A) \ne 0$ this inequality also doesn't hold. Consider:

$A=\left(\begin{array}{cc} 1 & a\\ 0 & 1 \end{array} \right)$

$det(A) = 1$ for every $a$

Let $x = \left(\begin{array}{cc} 1 \\ 1 \end{array} \right)$

Then: $Ax = \left(\begin{array}{cc} 1 \cdot 1 + a \cdot 1 \\ 0 \cdot 1 + 1 \cdot 1 \end{array} \right) = \left(\begin{array}{cc} a + 1 \\ 1 \end{array} \right)$

So clearly:

$\|Ax\| \to \infty$ when $a \to \infty$

but

$|det(A)|\|x\|=1 \cdot \sqrt 2$ for every $a$.