I have been given this set of three inequations (involving three unknowns):
$\underbrace{\left( \begin{array}{ccc} x-1 & -1 & -1 \\ -1 & y-1 & -1 \\ -1 & -1 & z-1 \end{array} \right)}_{\text{A}}\left( \begin{array}{c} a \\ b \\ c \end{array}\right) \geq \left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right)$
Given that inequality, the problem asks me to make an assumption on the determinant of the matrix (i.e. to find an equation that links $x$, $y$, and $z$).
I could solve the problem if there was an equality symbol (instead of a greater-or-equal-than symbol): it would be as simple as saying $det(A)=0$.
I could simply say that the condition is just $det(A) \geq 0$. Is this correct? If so, how can I get to that conclusion? A subtle hint is more than acceptable in order to help me solve this simple problem.