I know that the following statement is true but I cannot see how to prove it. I really appreciate if somebody could help. Thank you.
Let A be a 3x3, nonsingular matrix. Show that:
$$ [D^{-1} + P_3D^{-1}(L+U)P_{2}D^{-1}(L+U)P_{1}D^{-1}(L+U) - D^{-1}L]*D^{-1} = (D+L)^{-1} $$
where, $$P_1 = \begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix} \quad P_2 = \begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix} \quad P_3 = \begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}$$
and D, L, and U are the diagonal, lower triangular, and upper triangular matrices of matrix A, respectively.
UPDATE:
I simplified the original equation to: $$ DP_3D^{-1}LP_2D^{-1}LP_1D^{-1} = LD^{-1}LD^{-1} $$ by using properties of matrices and so on. I also verified it using numerical examples. However, I do not know how to further simplify the final equation. Any idea?