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

Question No. 33

In the LU decomposition of the matrix $\left[\begin{array}{cc}2& 2\\ 4& 9\end{array}\right]$,if the diagonal elements of $U$ are both 1, then the lower diagonal entry $l_{22}$ of $L$ is ______.

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

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