[Math] LU decomposition permutation matrix

Question

Hi can you help me with the following;

Let $A$ be an $n\times n$ matrix and have $LU$ decomposition
with lower and upper triangular matrices.
Let $P =\{e_n,e_{n-1},\ldots,e_1\}$ where $e_i$ is a unit vector i.e. $P$ is the permutation matrix.
Prove that $PAP$ has $UL$ factorization with $U$ upper triangular
and $L$ lower triangular matrix i.e. $PAP = UL$

I thought easily $PAP = (PL)(UP)$ are upper and lower parts but they are NOT!

Please help.

Thank you!!

Answer 1

Hint: $PAP = PLUP = (PLP)(PUP)$.