 Conjugating elimination matrices across a row permutation

For a permutation matrix $P$ and elimination matrices $E_i$, put $F_i=P{E}_i{P}^{-1}$. Cancellation of adjacent $P^{-1}P$ gives $P{E}_k\cdots E_{1}M=F_k\cdots F_1(PM)$. If the $F_i$ and their inverses are lower triangular, then $PM=LU$ with $U=P{E}_k\cdots E_{1}M$ and $L=F_1^{-1}\cdots F_k^{-1}$. Lower triangularity needs checking: an arbitrary permutation does not preserve it.