= Solution
Expand an arbitrary <vector> in the <orthonormal eigenbasis>:
$$
\mathbf r=r_-\mathbf e_-+r_0\mathbf e_0+r_+\mathbf e_+,
\qquad A\mathbf r=-6r_-\mathbf e_-+12r_+\mathbf e_+.
$$
Its image has no component along $\mathbf e_0$, so
$$
\boxed{(A\mathbf r)\cdot\mathbf e_0=0\quad\text{for every }\mathbf r,
\qquad\mathbf e_0=\frac1{\sqrt6}(1,2,1)^{\mathsf T}\ne0.}
$$
Alternatively, symmetry gives $(A\mathbf r)\cdot\mathbf e_0=\mathbf r\cdot A\mathbf e_0=0$. This is <image-kernel orthogonality for an adjoint>, with $A^{\mathsf T}=A$.
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