Show that if the rows of matrix can be split into two orthogonal linear subspaces, then the inverse has each pseudoinverse as columns

Question

Assume an invertible matrix $C$ is in the form $\left[ \begin{matrix}A \\ B \end{matrix}\right]$, where the linear subspace generated by the rows of $A$ is orthogonal to the linear subspace generated by the rows of $B$. Denote the Moore–Penrose pseudoinverses of $A$ and $B$ by $A^{+}$ and $B^{+}$, respectively.

How would you prove that $C^{-1}=\left[ \begin{matrix}A^{+} & B^{+} \end{matrix}\right]$?