For an $n$ by $n$ matrix $A$, find a factorization of the form $A = L\hat{U}$ where $L$ is a lower triangular matrix and for some permutation matrix, $P$ the matrix $P\hat{U}$ is upper triangular with 1's on the diagonal.
For example given matrix $A = $ \begin{bmatrix} 0&1&1\\ 0&-2&1\\ -2&1&1 \end{bmatrix}
I am able to get to this point:
$$A = L_5 P_5D^{-1}DU_5$$ where
$$L_5 = \begin{bmatrix} \frac{1}{2}&0&0\\-1&-2&0\\0&0&1 \end{bmatrix}$$ $$P_5 = \begin{bmatrix} 0&1&0\\0&0&1\\1&0&0 \end{bmatrix}$$ $$U_5 = \begin{bmatrix} -2&1&1\\0&2&2\\0&0&\frac{-3}{2} \end{bmatrix}$$
$$D = Diag\bigg(\frac{-1}{2}, \frac{1}{2}, \frac{-2}{3}\bigg)$$ $$D^{-1} = Diag\bigg(-2, 2, \frac{-3}{2}\bigg)$$
It took a lot of computation for me to even get to this point but I don't know how to proceed. I am not sure if there is a very simple clever way to do this.