Please solve the two questions. 4 and 5 Problem 4. Solve the tridiagonal system:Problem 5. Define the matrix An € Rn×n byAn2x1 + x₂ = 3-x1 + 2x₂ + x3 = 2-x₂ + 2x3 + x4 = 2−x3 +2x4+x5 = 2-x4 + 2x5=1=1 −1 -10-100⠀-1101. Show that the entry of the inverse of An at row i and column j has value:0,if i > j1,if i = J2j-i-1,if i < j1 2. What is the condition number of An? (The condition number of a matrix A is defined to be|| A||∞ || A¯¹||·) Is matrix An ill-conditioned?3. LetThe solution of Anx = bisPerturbĎ7 = (−n+2 −n+3toJ= (1 1b = b + (0What is â? (Hint: îLet â be the solution to Anî = 6.4. Show that the values of 7, 6, 7, î satisfy1)T.0 €) T.−1 0 1)¹.=À + Añ¹(ô − ¯').)||☎ − â||∞ ≤ || An||∞ || Añ¹² ||00|| 7² ||00n∞- bll00FL81100

“Since you have asked multiple questions, we will solve the first question for you. If you want any…

Answered: Solve the tridiagonal system: J C - 2x1…

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Solve the tridiagonal system: J C - 2x1 + x2 = -x1 + 2x2 + x3 = 2 -x2 + 2x3 + x4 = 2 -x3 +2x4+x5 = 2 -x4 + 2x5=1 = 3

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.8: Determinants

Problem 34E

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Question

100%

Please solve the two questions. 4 and 5

Problem 4. Solve the tridiagonal system:
Problem 5. Define the matrix An € Rn×n by
An
2x1 + x₂ = 3
-x1 + 2x₂ + x3 = 2
-x₂ + 2x3 + x4 = 2
−x3 +2x4+x5 = 2
-x4 + 2x5=1
=
1 −1 -1
0
-1
00
⠀
-1
1
0
1. Show that the entry of the inverse of An at row i and column j has value:
0,
if i > j
1,
if i = J
2j-i-1,
if i < j
1

2. What is the condition number of An? (The condition number of a matrix A is defined to be
|| A||∞ || A¯¹||·) Is matrix An ill-conditioned?
3. Let
The solution of Anx = bis
Perturb
Ď
7 = (−n+2 −n+3
to
J
= (1 1
b = b + (0
What is â? (Hint: î
Let â be the solution to Anî = 6.
4. Show that the values of 7, 6, 7, î satisfy
1)T.
0 €) T.
−1 0 1)¹.
=
À + Añ¹(ô − ¯').)
||☎ − â||∞ ≤ || An||∞ || Añ¹² ||00
|| 7² ||00
n
∞
- bll00
FL
81100

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