Consider solving the following system of simultaneous equations using LU…

GATE · 2022 · CS · Computer Science & IT

Consider solving the following system of simultaneous equations using LU decomposition.

x1+x2−2x3=4x1+3x2−x3=72x1+x2−5x3=7\begin{aligned}x_1+x_2-2x_3&=4\\x_1+3x_2-x_3&=7\\2x_1+x_2-5x_3&=7\end{aligned}

where L and U are denoted as

L=(L1100L21L220L31L32L33),U=(U11U12U130U22U2300U33)L=\begin{pmatrix}L_{11}&0&0\\L_{21}&L_{22}&0\\L_{31}&L_{32}&L_{33}\end{pmatrix},\quad U=\begin{pmatrix}U_{11}&U_{12}&U_{13}\\0&U_{22}&U_{23}\\0&0&U_{33}\end{pmatrix}

Which one of the following is the correct combination of values for L32L_{32}, U33U_{33}, and x1x_1?

  1. A.

    L32=2,U33=−12,x1=−1L_{32} = 2, U_{33} = - \frac {1}{2} , x_1 = -1

  2. B.

    L32=2,U33=2,x1=−1L_{32} = 2, U_{33} = 2 , x_1 = -1

  3. C.

    L32=−12,U33=2,x1=0L_{32} = - \frac {1}{2}, U_{33} = 2 , x_1 = 0

  4. D.

    L32=−12,U33=−12,x1=0L_{32} = - \frac {1}{2}, U_{33} = - \frac {1}{2} , x_1 = 0

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