Let $0,1,2$ is eigenvalues and $x_1,x_2,x_3$ is eigenvectors of matrix $A$
a) Decribe subspaces $N(A),R(A),R(A^T)$ in term of $x_1,x_2,x_3$
b) Find all solution $Ax=2x_2-3x_3$ in term $x_1,x_2,x_3$,what you can say about sistem $Ax=x_1+x_3$?
c)Check is matrix $A$ orthogonal
For a) $dimN(A)=1$ ( I do not get a dimension of matrix,but I think that is 3), $dimR(A)=2$ and $dimR(A^T)=2$, so $L(x_1)=N(A)$ and $L(x_2,x_3)=R(A)$ and $L(x_2,x_3)=R(A^T)$, I do not know is there something more to say. For b) I think that solution $x$ can show as linear combination of $x_2$,$x_3$, $x=2x_2-\frac{3}{2}x_3$ because $Ax=A(2x_2-\frac{3}{2}x_3)$=$2Ax_2-\frac{3}{2}Ax_3=2x_2-3x_3$, so $x=L((0,2,0),(0,0,-3))$, for other part I think that $Ax=x_1+x_3$ have no solution because $x_1$ is solution for $Ax=0$ and $x\in R(A)$ so it can only show as linear combination of vectors that span $R(A)$ and that is $x_2,x_3$ , and for c) $A$ is not orthogonal since matrix $A$ does not have full rank, is this ok?
For $(a)$ I'm not sure how you determined $R(A^T)$. You can only say $$R(A^T) = N(A)^\perp = L(x_1)^\perp$$
For $(b)$ we have
$$Ax = 2x_2-3x_3 = A\left(2x_2 - \frac32x_3\right) \implies x - 2x_2 + \frac32x_3 \in N(A)$$
so $x = 2x_2 - \frac32x_3 + \lambda x_1$ for some $\lambda \in \mathbb{R}$.
The other equation $Ax = x_1 + x_3$ has no solutions because $x_1 + x_3 \notin L(x_2, x_3) = R(A)$.
For $(c)$, another argument that the matrix is not orthogonal is that the eigenvalues are not contained in the unit circle.