# Equation - Basic operations on matrices

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):

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

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Remember that $(AB)^{-1} = B^{-1}*A^{-1}$ and use this on $A = D$ and $B = I + D^{-1}L$ –  Cocopuffs Aug 18 '12 at 17:19
Perfect, thank you! –  Chris Aug 18 '12 at 20:55

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