Solution
= Solution
The columnwise <null space> conditions say $A^\top L=0$ and $AR=0$. Therefore
$$
\boxed{A^\top B=A^\top LR^\top=0,\qquad
AB^\top=A(RL^\top)=(AR)L^\top=0.}
$$
Both right-hand sides are the $n\times n$ <zero matrix>.
= Solution
The columnwise <null space> conditions say $A^\top L=0$ and $AR=0$. Therefore
$$
\boxed{A^\top B=A^\top LR^\top=0,\qquad
AB^\top=A(RL^\top)=(AR)L^\top=0.}
$$
Both right-hand sides are the $n\times n$ <zero matrix>.