Equation - Basic operations on matrices

Question

Let $x^{(k)}$ and $b$ be vectors and $L,\ D, \ U$ be lower, diagonal, upper triangular matrices.

We have:

$x^{(k)}=(I+D^{-1}L)^{-1}(D^{-1}b-D^{-1}Ux^{(k-1)})$ (1)

How does the following follow from (1):

$x^{(k)}=-(D+L)^{-1}Ux^{(k-1)}+(D+L)^{-1}b$

Answer 1

With Cocopuffs input:

$(I+D^{-1}L)^{-1}(D^{-1}b-D^{-1}Ux^{(k-1)})=(I+D^{-1}L)^{-1}D^{-1}(b-Ux^{(k-1)})=(D(I+D^{-1}L))^{-1}(b-Ux^{(k-1)})=-(D+L)^{-1}Ux^{(k-1)}+(D+L)^{-1}b$