<s>[INST] This problem involves determinant equations. Suppose we have some constant matrix $M$. The equation is as follows:

$$\text{det}(Mx) = \text{det}(My)$$

Where $x, y$ are some column vectors and $Mx, My$ are their product with the matrix $M$. Does $\text{det}(Mx) = \text{det}(My)$ imply $x=y$?[/INST]