Please don't provide handwritten solution ...... 5. Let302 145-4 4 -7 1Find a PLU-factorization of A, and use it to solve the linear system Az = b.0-21 -12-2A2169-20

We have, Start from identity matrix and identity permutation matrix Here, L is the multipliers used…

Answered: 5. Let 0-2 1 -1 2-2 30 21 45 -4 4 -7 1…

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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Please don't provide handwritten solution ......

5. Let
30
2 1
45
-4 4 -7 1
Find a PLU-factorization of A, and use it to solve the linear system Az = b.
0-2
1 -1
2-2
A
21
6
9
-20

Expert Solution

Step 1: Find the PLU factorization of A

We have,

$A equals open square brackets table row 0 cell negative 2 end cell 3 0 row 1 cell negative 1 end cell 2 1 row 2 cell negative 2 end cell 4 5 row cell negative 4 end cell 4 cell negative 7 end cell 1 end table close square brackets$

Start from identity matrix $L equals open square brackets table row 1 0 0 0 row 0 1 0 0 row 0 0 1 0 row 0 0 0 1 end table close square brackets$ and identity permutation matrix $P equals open square brackets table row 1 0 0 0 row 0 1 0 0 row 0 0 1 0 row 0 0 0 1 end table close square brackets$

$A p p l y i n g space R subscript 1 left right arrow R subscript 2 open square brackets table row 1 cell negative 1 end cell 2 1 row 0 cell negative 2 end cell 3 0 row 2 cell negative 2 end cell 4 5 row cell negative 4 end cell 4 cell negative 7 end cell 1 end table close square brackets space space space space space S o comma space P equals open square brackets table row 0 1 0 0 row 1 0 0 0 row 0 0 1 0 row 0 0 0 1 end table close square brackets R subscript 3 rightwards arrow R subscript 3 minus 2 cross times R subscript 1 A equals open square brackets table row 1 cell negative 1 end cell 2 1 row 0 cell negative 2 end cell 3 0 row 0 0 0 3 row cell negative 4 end cell 4 cell negative 7 end cell 1 end table close square brackets space space space space S o comma space L equals open square brackets table row 1 0 0 0 row 0 1 0 0 row 2 0 1 0 row 0 0 0 1 end table close square brackets R subscript 4 rightwards arrow R subscript 4 minus open parentheses negative 4 close parentheses cross times R subscript 1 open square brackets table row 1 cell negative 1 end cell 2 1 row 0 cell negative 2 end cell 3 0 row 0 0 0 3 row 0 0 1 5 end table close square brackets space space space space space S o comma space L equals open square brackets table row 1 0 0 0 row 0 1 0 0 row 2 0 1 0 row cell negative 4 end cell 0 0 1 end table close square brackets R subscript 3 left right arrow R subscript 4 U equals open square brackets table row 1 cell negative 1 end cell 2 1 row 0 cell negative 2 end cell 3 0 row 0 0 1 5 row 0 0 0 3 end table close square brackets space space space space space S o comma space P equals open square brackets table row 0 1 0 0 row 1 0 0 0 row 0 0 0 1 row 0 0 1 0 end table close square brackets space a n d space L equals open square brackets table row 1 0 0 0 row 0 1 0 0 row cell negative 4 end cell 0 1 0 row 2 0 0 1 end table close square brackets T h e r e f o r e space P L U thin space f a c t o r i z a t i o n space o f space A space i s space P A equals L U$

Here, L is the multipliers used in the elimination with 1 on its diagonal.