$D^-$ of the adjugate of $X$: $D^-(\operatorname{adj}X) = -(\operatorname{adj}X)\,A$

Let $X(r,\theta)$ be the matrix integrating factor of Hintz eq. (5) and $A$ the Hermitian coefficient matrix of eq. (4), on the domain $U = \{r > r_+,\ 0<\theta<\pi\}$ of a subextremal Kerr–Newman background ($M>0$, $Q\neq 0$, $a^2+Q^2<M^2$). Since $\det X = Q^2 - M^2$ is a nonzero constant on $U$, the adjugate satisfies $\operatorname{adj}X = (Q^2-M^2)\,X^{-1}$ there, and the integrating-factor equation $D^-X = AX$ transports to

where $D^- = \sqrt{\Delta}\,\partial_r - i\,\partial_\theta$ acts entrywise. This is the derivative rule needed for the transformed variable $z_+ = (\operatorname{adj}X)\,y_+$ of eq. (7).