= Solution
Write the horizontal velocity amplitudes as $(u,v)$ and magnetic amplitudes as $(b_x,b_y)$. The perturbation is horizontally uniform, divergence-free, and has no vertical velocity, so the unperturbed density and pressure are consistent at linear order. In the <Keplerian shearing sheet>, the horizontal components of the <linearized ideal magnetohydrodynamic equations> are
$$
su-2\Omega v=\frac{v_A^2}{h}\frac{F''}{ikF}b_x,\qquad
sv+\frac\Omega2u=\frac{v_A^2}{h}\frac{F''}{ikF}b_y.
$$
Use the undivided equations at a zero of $F$. If $F''=-k^2hF$, the magnetic-force coefficient becomes $ikv_A^2$; the stratification cancels. The <ideal magnetohydrodynamic induction equation> similarly reduces to
$$
sb_x=iku,\qquad sb_y=-\frac32\Omega b_x+ikv.
$$
The azimuthal induction term is field stretching by the background <differential rotation>. The four amplitudes obey the homogeneous system
$$
\begin{pmatrix}
s&-2\Omega&-ikv_A^2&0\\
\Omega/2&s&0&-ikv_A^2\\
-ik&0&s&0\\
0&-ik&3\Omega/2&s
\end{pmatrix}
\begin{pmatrix}u\\v\\b_x\\b_y\end{pmatrix}=0.
$$
Its <determinant> must vanish for a nonzero <normal mode>. Put $A=v_A^2k^2$. For nonzero $s$, elimination first gives $(s^2+A)u-2\Omega sv=0$ and $s(s^2+A)v+\Omega(s^2-3A)u/2=0$, whose solvability condition is $(s^2+A)^2+\Omega^2(s^2-3A)=0$. The original determinant extends the same polynomial to marginal $s=0$. Thus
$$
\boxed{s^4+(\Omega^2+2v_A^2k^2)s^2+v_A^2k^2(v_A^2k^2-3\Omega^2)=0.}
$$
This is the <ideal magnetorotational dispersion relation> with the midplane <Alfvén speed> $v_A=B_0/\sqrt{4\pi\rho_0}$. The vertical structure enters through the admissible eigenvalues $k$, rather than through a different horizontal dispersion polynomial. No division by a vanishing growth rate is required in the determinant derivation.
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