I am currently working on a computationally expensive matrix multiplication question that I keep getting hung up on and would like further direction as to where I am making a mistake. Note that the matrices $L, U, D$ are the splitting of some matrix $A$ ($i.e., A = L + U + D$), but that's not too important for my question:
Given that $A_{1}=(D+L)D^{-1}(D+U)$, $A_{2}=-LD^{-1}U$, $M=-A_{1}^{-1}A_{2}$, and $M_{\text{SGS}}=(D+U)^{-1}L(D+L)^{-1}U$ show that $M^{-1}=M^{-1}_{SGS}.$
Here is what I have tried and where I am getting stuck:
Start with noting that $M^{-1}_{SGS}=U^{-1}(D+L)L^{-1}(D+U)$, now computing $M^{-1}$ and expanding we get that
\begin{align*} M^{-1} &= -[-U^{-1}DL^{-1}][(D+L)D^{-1}(D+U)] \\ &= [U^{-1}DL^{-1}][D+U+L+LD^{-1}U] \\ &=U^{-1}DL^{-1}D + U^{-1}DL^{-1}U + U^{-1}DL^{-1}L + U^{-1}DL^{-1}LD^{-1}U \\ &=U^{-1}DL^{-1}D + U^{-1}DL^{-1}U + U^{-1}D + I \end{align*}
Sadly, this is where I keep getting stuck. I'm not sure what to do with the identity matrix term at the end of the last line which makes me believe I should not approach the distributing of terms this way. additionally, I don't even know how I would collect the factors $(D+L)$ or $(D+U)$ at this point to piece together $M^{-1}_{\text{SGS}}$.
Any help is appreciated.
In case the reader was wondering, the matrix $M_{\text{SGS}}$ denotes a symmetric Gauss-Seidel iteration matrix. This does not imply any of the matrices given are symmetric.
You were quite close actually. Just factorize all the terms as follows \begin{align*} M_{\text SGS}^{-1} &= U^{-1} (D + L) L^{-1} (D + U) = (U^{-1} D + U^{-1} L) (L^{-1} D + L^{-1} U) \\ &= U^{-1} D L^{-1} D + U^{-1} D L^{-1} U + U^{-1} L L^{-1} D + U^{-1} L L^{-1} U \\ &= U^{-1} D L^{-1} D + U^{-1} D L^{-1} U + U^{-1} D + I \\ &= M^{-1} \end{align*} and you are done :-)