2. Find the LU factorizations of\[\begin{bmatrix}2 & 0 & 5 & 2 \\-6 & 3 & -13 & -3 \\4 & 9 & 16 & 17\end{bmatrix}\]and\[\begin{bmatrix}2 & 3 & 2 \\4 & 13 & 9 \\-6 & 5 & 4\end{bmatrix}\]

First, we have 2052-63-13-3491617 Let the lower triangular matrix be 100L2,110L3,1L3,21 Consider…

Answered: 2. Find the LU factorizations of and 20…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

2. Find the LU factorizations of

\[
\begin{bmatrix}
2 & 0 & 5 & 2 \\
-6 & 3 & -13 & -3 \\
4 & 9 & 16 & 17
\end{bmatrix}
\]

and

\[
\begin{bmatrix}
2 & 3 & 2 \\
4 & 13 & 9 \\
-6 & 5 & 4
\end{bmatrix}
\]

Expert Solution

Step 1

First, we have $[\begin{matrix} 2 & 0 & 5 & 2 \\ - 6 & 3 & - 13 & - 3 \\ 4 & 9 & 16 & 17 \end{matrix}]$

Let the lower triangular matrix be $[\begin{matrix} 1 & 0 & 0 \\ L_{2, 1} & 1 & 0 \\ L_{3, 1} & L_{3, 2} & 1 \end{matrix}]$

Consider $[\begin{matrix} 2 & 0 & 5 & 2 \\ - 6 & 3 & - 13 & - 3 \\ 4 & 9 & 16 & 17 \end{matrix}]$

Apply $R_{2} \to R_{2} + 3 R_{1}$ (which gives $L_{2, 1} = - 3$ )

$[\begin{matrix} 2 & 0 & 5 & 2 \\ 0 & 3 & 2 & 3 \\ 4 & 9 & 16 & 17 \end{matrix}]$

Apply $R_{3} \to R_{3} - 2 R_{1}$ (which gives $L_{3, 1} = 2$ )

$[\begin{matrix} 2 & 0 & 5 & 2 \\ 0 & 3 & 2 & 3 \\ 0 & 9 & 6 & 13 \end{matrix}]$

Apply $R_{3} \to R_{3} - 3 R_{2}$ (which gives $L_{3, 2} = 3$ )

$[\begin{matrix} 2 & 0 & 5 & 2 \\ 0 & 3 & 2 & 3 \\ 0 & 0 & 0 & 4 \end{matrix}]$

So, the upper triangular matrix is $[\begin{matrix} 2 & 0 & 5 & 2 \\ 0 & 3 & 2 & 3 \\ 0 & 0 & 0 & 4 \end{matrix}]$ and the lower triangular matrix is $[\begin{matrix} 1 & 0 & 0 \\ - 3 & 1 & 0 \\ 2 & 3 & 1 \end{matrix}]$

So, $L U$ decomposition of $[\begin{matrix} 2 & 0 & 5 & 2 \\ - 6 & 3 & - 13 & - 3 \\ 4 & 9 & 16 & 17 \end{matrix}]$ is :

$[\begin{matrix} 2 & 0 & 5 & 2 \\ - 6 & 3 & - 13 & - 3 \\ 4 & 9 & 16 & 17 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ - 3 & 1 & 0 \\ 2 & 3 & 1 \end{matrix}] [\begin{matrix} 2 & 0 & 5 & 2 \\ 0 & 3 & 2 & 3 \\ 0 & 0 & 0 & 4 \end{matrix}]$