= Solution
The $k^{-1}$ coefficient of the first equation is $\mu_{1x}+i[\sigma,\mu_2]=Q\mu_1$. Substitution in the expression for $B$ gives $B=-2\mu_{1x}$. Decompose $\mu_1$ into diagonal and off-diagonal parts. Since $Q=i[\sigma,\mu_1]$, its off-diagonal part is $iQ\sigma/2$. Thus
$$
(\mu_{1x})_{\mathrm{off}}=\tfrac i2Q_x\sigma,\qquad
(\mu_{1x})_{\mathrm{diag}}=(Q\mu_1)_{\mathrm{diag}}=\tfrac i2Q^2\sigma.
$$
The commutator in the $k^{-1}$ equation has zero diagonal; multiplication of two off-diagonal two-by-two <matrices> is diagonal. Combining the two parts proves
$$
\boxed{A=2Q,\qquad B=-i(Q^2+Q_x)\sigma_3.}
$$
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