GATE Papers >> CSE >> 2015 >> Question No 33

Question No. 33 CSE | GATE 2015

In the LU decomposition of the matrix 2249,if the diagonal elements of $U$ are both 1, then the lower diagonal entry $l_{22}$ of $L$ is ______.

Answer : 5

Solution of Question No 33 of GATE 2015 CSE Paper

$ \begin{bmatrix}L_{11}&0\\L_{21}&L_{22}\end{bmatrix}\begin{bmatrix}1&U_{12}\\0&1\end{bmatrix}=\begin{bmatrix}2&2\\4&9\end{bmatrix} $

This is crout's LU decomposition, since diagonal elements of U are 1. So we will setup the equations for the elements of the matrix taken column-wise, as follows

$ \begin{array}{l}\;\;\;\;\;L_{11}=2,\;L_{21}=4\\\;\;\;\;\;\;\;\;\;\;L_{11}\times U_{12}=2\\\Rightarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;U_{12}=1,\\L_{21}\times U_{12}+L_{22}=9\\\Rightarrow\;\;\;\;\;\;\;\;4+{\mathrm L}_{22}=9\\\Rightarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;L_{22}=5\end{array} $

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