Official PYQ · 2015Set 1 · Q1Linear Algebra: Matrices, Determinants, System of Linear EquationsNAT1 mark

GATE CS 2015 Set 1 Q1 — Linear Algebra: Matrices, Determinants, System of Linear Equations

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}

L

is_________________.

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Answer: 5

We are given the matrix $A = \begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$ and its $LU$ decomposition, where the diagonal elements of $U$ are both $1$ . Let:

L = \begin{bmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{bmatrix} \quad \text{and} \quad U = \begin{bmatrix} 1 & u_{12} \\ 0 & 1 \end{bmatrix}

Multiplying $L$ and $U$ , we get:

LU = \begin{bmatrix} l_{11} & l_{11} u_{12} \\ l_{21} & l_{21} u_{12} + l_{22} \end{bmatrix}

By equating $LU = A$ , we compare the corresponding elements:

$l_{11} = 2$
$l_{11} u_{12} = 2 \implies 2 u_{12} = 2 \implies u_{12} = 1$
$l_{21} = 4$
$l_{21} u_{12} + l_{22} = 9 \implies 4(1) + l_{22} = 9 \implies l_{22} = 5$

Thus, the lower diagonal entry $l_{22}$ of $L$ is 5.

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Published 10 Oct 2026

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