Consider the permutation matrix
$P= \begin{pmatrix} 0 & \cdots & 0 & 1 \\ 0 & \cdots & 1 &0 \\ \vdots & \ddots & \vdots & \vdots \\ 1 & \cdots & 0 & 0 {} \end{pmatrix}$
SoI showed that if $L$ is lower triangular then $PLP$ is upper triangular.
Now the problem asks me to descrive how to calculate a factorisation $A=\overline U \ \overline L $
where $\overline U$ is unit upper triangular and $\overline L$ is lower triangular.
I am finding it difficult because I think the previous part was an hint to do the following:
$PAP=PLUP=PLP^2UP$ as $P^2=I$
And then simply define $\overline U = PLP$ and $\overline P = PUP$
How can I get rid of the $P$ s on the rhs?
Any hint would be appreciated, I think I am missing something!