Given a $n$-x-$d$ matrix (with $n>d$) $A$ whose columns are linearly independent, does there exist another $n$-x-$d$ matrix $B$ that satisfies both
$$B'A=I_d$$
(where $I_d$ is a $d$-x-$d$ identity matrix)
and
$$BA'=\begin{bmatrix} 1 & 0 &0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
Assume both $A$ and $B$ are full rank with non-zero real valued entries.