Let $A \in \mathbb{K}^{n \times n}$ a matrix, and suppose that we can run the Gaussian elimination on $A$ without row or column interchange, so there exist the $LU$ decomposition of $A$.
We define the $LU$ decomposition as $U \in \mathbb{K}^{n \times n}$ is an upper triangle matrix, and $L \in \mathbb{K}^{n \times n}$ is a lower triangle matrix, with $\ell_{i,i}=1$ main diagonal elements, and $A=LU$.
We know, that if $\det(A) \neq 0$, so $A$ is invertible, then the $LU$ decomposition is unique. Quite easy to show that by contradiction.
Questions.
$(1)$ Is there any other condition for the uniqueness of $LU$ decomposition, even if $\det(A)=0$?
$(2)$ Is there a necessary and sufficient condition for uniqueness of $LU$ decomposition?