Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 17E from Chapter 6.SE from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 17E
Chapter:
Problem:
0
Step-by-Step Solution
Given Information
We are given with following matrices:\[A=\left[\begin{array}{cc}{4.5} & {3.1} \\ {1.6} & {1.1}\end{array}\right], \mathbf{b}=\left[\begin{array}{c}{19.249} \\ {6.843}\end{array}\right], \Delta \mathbf{b}=\left[\begin{array}{c}{0.001} \\ {-0.003}\end{array}\right]\]We have to solve the systems $A \mathbf{x}=\mathbf{b} \text { and } A(\Delta \mathbf{x})=\Delta \mathbf{b}$
Step-1:
Enter the matrices in MATLAB:
>> A=[4.5 3.1 ; 1.6 1.1]
>> b=[19.249; 6.843]
>> deltab=[0.001;-0.003]
Step-2:
Solve the system of equation Ax=b:
>> x=A\b
x =
3.9400
0.4900
Solve the system of equation A(deltax)=deltab:
>> >> deltax=A\deltab
deltax =
-1.0400
1.5100
Step-3:
Find the norm and condition number:
>> norm(deltax)/norm(x)
ans =
0.4618
>> norm(deltab)/norm(b)
ans =
1.5479e-04
Condition Number of A:
>> cond(A)
ans =
3.3630e+03
Step-4:
Compute the ratio $\operatorname{cond}(A) \cdot \dfrac{\|\Delta \mathbf{b}\|}{\|\mathbf{b}\|}$
>> cond(A)* norm(deltab)/norm(b)
ans =
0.5206
Therefore,
\[\dfrac{\|\Delta \mathbf{x}\|}{\|\mathbf{x}\|} \leq \operatorname{cond}(A) \cdot \dfrac{\|\Delta \mathbf{b}\|}{\|\mathbf{b}\|}\]
We are given with following matrices:\[A=\left[\begin{array}{cc}{4.5} & {3.1} \\ {1.6} & {1.1}\end{array}\right], \mathbf{b}=\left[\begin{array}{c}{19.249} \\ {6.843}\end{array}\right], \Delta \mathbf{b}=\left[\begin{array}{c}{0.001} \\ {-0.003}\end{array}\right]\]We have to solve the systems $A \mathbf{x}=\mathbf{b} \text { and } A(\Delta \mathbf{x})=\Delta \mathbf{b}$
Step-1:
Enter the matrices in MATLAB:
>> A=[4.5 3.1 ; 1.6 1.1]
>> b=[19.249; 6.843]
>> deltab=[0.001;-0.003]
Step-2:
Solve the system of equation Ax=b:
>> x=A\b
x =
3.9400
0.4900
Solve the system of equation A(deltax)=deltab:
>> >> deltax=A\deltab
deltax =
-1.0400
1.5100
Step-3:
Find the norm and condition number:
>> norm(deltax)/norm(x)
ans =
0.4618
>> norm(deltab)/norm(b)
ans =
1.5479e-04
Condition Number of A:
>> cond(A)
ans =
3.3630e+03
Step-4:
Compute the ratio $\operatorname{cond}(A) \cdot \dfrac{\|\Delta \mathbf{b}\|}{\|\mathbf{b}\|}$
>> cond(A)* norm(deltab)/norm(b)
ans =
0.5206
Therefore,
\[\dfrac{\|\Delta \mathbf{x}\|}{\|\mathbf{x}\|} \leq \operatorname{cond}(A) \cdot \dfrac{\|\Delta \mathbf{b}\|}{\|\mathbf{b}\|}\]