Suppose $a_{0},a_{1},a_{2}\in\mathbb{Q}$, such that the following determinant is zero, i.e.

$ \left |\begin{array}{cccc}\\ a_{0} &a_{1} & a_{2} \\ \\ a_{2} &a_{0}+a_{1} & a_{1}+a_{2} \\ \\ a_{1} &a_{2} & a_{0}+a_{1}\\ \end{array}\right| =0$

Problem.Show that $a_{0}=a_{1}=a_{2}=0.$

I think it's equivalent to show that the rank of the matrix is $0$, and it's easy to show the rank cannot be $1$. But I have no idea how to show that the case of rank 2 is impossible. So, is there any better idea?

1more comment