the solution set is a subspace generated by .
Both second and third equations just tell you that x= 0. The first equation tells you that 2y= -3z so y= -(3/2)z. Any eigenvector is of the form <0, -(3/2)z, z>= z<0, -3/2, 1>. If z= 2, that is <0, -3 , 2> so the space of eigenvectors of this matrix is spanned by that single vector (which is, of course, a multiple of the one Shanks gave).