question:find all real numbers λ such that the homogenous system

(λIn-A)x=0has a nontrivial solution

$\displaystyle A=\left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & -1 \\ 1 & 0 & 0 \end{array} \right]$

work:$\displaystyle (\Lambda I_n - A)=\left[\begin{array}{ccc} \Lambda & 0 & 0 \\ 0 & \Lambda-1 & -1 \\ 1 & 0 & \Lambda \end{array} \right]$. Then the answer key says that (λIn-A)x=0has a nontrivial solution if and only if its determinant=0. So then to solve the problem the determinant is taken and the values for λ are used such that the determinant is 0. I'm sceptical about setting the determinant to 0 because if a system has a determinant that is 0 then it means the system only has the trivial solution or infinetly many solutions. So couldn't this mean that the system only has trivial solutions? or am I lossing sight of the question being asked