 Solution

Eliminate the first column using multipliers $3$ and $2$. The two residual rows have second entries zero and four, so interchange their order for the next pivot. One resulting Pivoted LU decomposition isThe factors below the first pivot must follow the permuted row order, which is why $L$ contains $2$ before $3$. Direct multiplication gives rows $(3,1,4)$, $(6,6,6)$ and $(9,3,1)$ in both $PA$ and $LU$. This checks the permutation convention rather than confusing $PA=LU$ with a different placement of $P$.