Question

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

Answer 1

A=LU
$\Rightarrow \left[\begin{array}{cc}2& 2\\ 4& 9\end{array}\right]$=$\left[\begin{array}{cc}{l}_{11}& 0\\ l_{21}& l_{22}\end{array}\right]\left[\begin{array}{cc}1& u_{12}\\ 0& 1\end{array}\right]$
Comparing the correspondig elements on the both sides,
2=$l_{11}$
(2=l_{11}.u_{12}\Rightarrow u_{12}=1\)
$4=l_{21}$
$9=l_{21}.u_{12}+l_{22}\Rightarrow 9=4\times l+l_{22}$
$\Rightarrow l_{22}=5$