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Question A.10: Transform the following matrix into LU form |1 2 1 0 0 3 3 1......

Transform the following matrix into LU form

\left|\begin{matrix}1 &2 &1 &0 \\ 0& 3& 3& 1 \\ 2& 0& 2& 0 \\  1 &0 &0 &2 \end{matrix} \right|
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From Equations A.75 and A.76,

l_{ij}=a_{ij}-\sum\limits_{k=1}^{k=j-1}{l_{ik}u_{kj}} \ \ i\geq j    (A.75)
u_{ij}=\frac{1}{l_{ij}}(a_{ij}-\sum\limits_{k=1}^{k=j-1}{l_{ik}u_{kj}}) \ \ i< j    (A.76)

\left|\begin{matrix}1 &2 &1 &0 \\ 0& 3& 3& 1 \\ 2& 0& 2& 0 \\  1 &0 &0 &2 \end{matrix} \right| = \left|\begin{matrix}1 &0 & 0&0 \\ 0& 3& 0& 0 \\ 2& -4& 4& 0 \\  1 &-2 &1 &2.33 \end{matrix} \right|\left|\begin{matrix}1 &2 &1 &0 \\ 0& 1& 1& 0.33 \\ 0& 0& 1& 0.33 \\  0 &0 &0 &1 \end{matrix} \right|

The original matrix has been converted into a product of lower and upper triangular matrices.

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