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GATE 2022 CS – Question 45

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Consider solving the following system of simultaneous equations using LU decomposition.
$x_1+x_2-2x_3=4$
$x_1+3x_2-x_3=7$
$2x_1+x_2-5x_3=7$
where $L$ and $U$ are denoted as
$L=\begin{pmatrix}L_{11}&0&0\\L_{21}&L_{22}&0\\L_{31}&L_{32}&L_{33}\end{pmatrix}$, $U=\begin{pmatrix}U_{11}&U_{12}&U_{13}\\0&U_{22}&U_{23}\\0&0&U_{33}\end{pmatrix}$
Which one of the following is the correct combination of values for $L_{32}$, $U_{33}$, and $x_1$?

  1. $L_{32}=2,\ U_{33}=-\frac12,\ x_1=-1$
  2. $L_{32}=2,\ U_{33}=2,\ x_1=-1$
  3. $L_{32}=-\frac12,\ U_{33}=2,\ x_1=0$
  4. $L_{32}=-\frac12,\ U_{33}=-\frac12,\ x_1=0$

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Correct answer: (D) $L_{32}=-\frac12,\ U_{33}=-\frac12,\ x_1=0$

Explanation

Doolittle factorization gives $L_{21}=1$, $L_{31}=2$, $U_{22}=2$, $U_{23}=1$, $L_{32}=(1-2)/2=-\frac12$ and $U_{33}=-5+4+\frac12=-\frac12$. Forward substitution gives $y=(4,3,\frac12)$. Back substitution gives $x_3=-1$, $x_2=2$, $x_1=0$.