Given a $n \times n$ matrix $A$, when do the first $p$ columns of $A^{-1}$ coincide with the Moore–Penrose inverse of the first $p$ rows of $A$?

Question

I noticed that for

${A}= \left[\begin{matrix} \cos{\left(\phi\right)} & \cos{\left(\phi+2\pi/3\right)} & \cos{\left(\phi+4\pi/3\right)}\\ \sin{\left(\phi\right)} & \sin{\left(\phi+2\pi/3\right)} & \sin{\left(\phi+4\pi/3\right)}\\ \beta & \beta & \beta \end{matrix}\right]$ and $p=2$ the proposition in the title holds $\forall \phi$.

I wonder if there is any generalization to this fact, i.e. the minimal structure that a generic matrix $A$ needs to have in order to satisfy $$(PA)^{\dagger}={A}^{-1}P^T$$ where $P$ is a projection of the form $\left[\begin{matrix}I &0\end{matrix}\right]$ with $I$ the identity matrix of size $p$ and $0$ a zero matrix of size $p \times n$.

Answer 1

It happens when the first $p$ rows are orthogonal to the other ones.