Say I would like to swap the two rows, $i$th and $j$th in the matrix $A$. Then the appropriate permutation matrix is
$$ P_{ij}^T = \left[e_1 \cdots e_{i-1} e_j e_{i+1} \cdots e_{j-1} e_i e_{j+1} \cdots e_n \right] $$
where WLOG $i < j$ assumed,
which can be premultiplied to $A$ such that
$$ \tilde{A}_{ij} = P_{ij} A $$
where $\tilde{A}_{ij}$ denotes the $i,j$-column swapped version of $A$.
Question
The above reasoning is quite straightforward to me, but I would like to know when the permutation matrix is unique as $P_{ij}$.
The permutation matrix is not unique, because When $A=0$ any row permutation of $A$ can be denoted as follows:
$$ PA = A $$
where $P$ can be, any matrix with the appropriate size.
So I'm wondering if there is any necessary/sufficient condition for the $P_{ij}$ to be the unique row exchanging matrix. Any help will be appreciated.
Edited
From a clever comment, I realized that
$$ P_{ij} = \tilde{A_{ij}} A^{-1} $$
if $A^{-1}$ exists. However, I would like to know if this uniqueness is preserved even when $A^{-1}$ does not exist.