GATE 2015 CS – Question 33
In the LU decomposition of the matrix $\begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$, if the diagonal elements of $U$ are both 1, then the lower diagonal entry $l_{22}$ of $L$ is ________.
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Correct answer: 5
Explanation
Write $L = \begin{bmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{bmatrix}$ and $U = \begin{bmatrix} 1 & u_{12} \\ 0 & 1 \end{bmatrix}$. Multiplying gives $l_{11} = 2$, $l_{11}u_{12} = 2$ so $u_{12} = 1$, $l_{21} = 4$, and $l_{21}u_{12} + l_{22} = 9$, so $4 + l_{22} = 9$ and $l_{22} = 5$.