![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_largeCoverImage.gif)
(a) Solve the following system of equations by LU decomposition without pivoting
(b) Determine the matrix inverse. Check your results by verifying that
(a)
![Check Mark](/static/check-mark.png)
To calculate: The solution of the system of equations given below by LU decomposition without pivoting.
Answer to Problem 3P
Solution:
The solution of the system of equations is
Explanation of Solution
Given:
The system of equations,
Formula used:
(1) The forward substitution equations for L can be expressed as,
(2) The backward substitution equation for U can be expressed as,
Calculation:
Consider the system of equations,
The coefficient
And subtracting the result from equation (2).
Thus, multiply equation (1) by
Now subtract this equation from equation (2),
The coefficient
And subtracting the result from equation (3).
Thus, multiply equation (1) by
Now subtract this equation from equation (3),
Now the set of equations is,
The factors
The coefficient
And subtracting the result from equation (5). Thus, multiply equation (4) by
Now, subtract this equation from equation (5),
The factor
Therefore, the LU decomposition is
Now, to find the solution of the given system:
The forward substitution equations for L can be expressed as,
Solve for
Solve for
Solve for
Thus,
Now, perform backward substitution:
Solve for
Solve for
Solve for
Thus,
(b)
![Check Mark](/static/check-mark.png)
To calculate: The matrix inverse for given system of equations and check the result by verifying that
Answer to Problem 3P
Solution:
The matrix inverse is
Explanation of Solution
Given:
The system of equations,
And the LU decomposition is
Formula used:
(1) The forward substitution equations for L can be expressed as,
(2) The backward substitution equation for U can be expressed as,
Calculation:
Consider the given system of equations:
The matrix [A] is:
The lower and upper triangular matrix after decomposition are given as:
The first column of the inverse matrix can be determined by performing the forward substitution solution with a unit vector (with 1 in the first row) of right-hand-side vector.
The forward substitution equations for L can be expressed as,
Where,
Determine D by substituting L and B as shown below,
Solve for
Solve for
Solve for
Hence, the values obtained are
Solve with forward substitution of
This vector can be used as right-hand side vector of equation,
Solve the above matrix by back substitution, which gives the first column of the inverse matrix as:
Similarly, the second column of the inverse matrix can be determined by performing the forward substitution solution with a unit vector (with 1 in the second row) of right-hand-side vector.
The forward substitution equations for L can be expressed as,
Where,
Determine D by substituting L and B as shown below,
Solve for
Solve for
Solve for
Hence, the values obtained are
Solve with forward substitution of
This vector can be used as right-hand side vector of equation,
Solve the above matrix by back substitution, which gives the second column of the inverse matrix as:
Similarly, the third column of the inverse matrix can be determined by performing the forward substitution solution with a unit vector (with 1 in the third row) of right-hand-side vector.
The forward substitution equations for L can be expressed as,
Where,
Determine D by substituting L and B as shown below,
Solve for
Solve for
Solve for
Hence, the values obtained are
Solve with forward substitution of
This vector can be used as right-hand side vector of equation,
Solve the above matrix by back substitution, which gives the third column of the inverse matrix as:
Thus, the inverse matrix is:
Now, check the result obtained.
Hence, verified.
Want to see more full solutions like this?
Chapter 10 Solutions
Numerical Methods for Engineers
Additional Math Textbook Solutions
Pathways To Math Literacy (looseleaf)
College Algebra (Collegiate Math)
Precalculus
Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
Intermediate Algebra (13th Edition)
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
- A 315-N force F and 70-N-m couple M are applied to co rner A of the block shown. Replace the given force -couple system with an equivalent force -couple systemat corner Darrow_forwardA force and couple are applied to a beam as shown below . (a) Replace this systemwith a single force F applied at point G, and determine the distance d. (b) Solve part aassuming that the directions of the two 600 -N forces are reversed Help finding the distancesarrow_forwardx If f(x) = = L* f(t)dt then find the value of ƒ (ln 7). f: R R is continuous everywhere.arrow_forward
- An eccentric, compressive 270-lb force P is applied to the end of a cantilever beam. Replace P with an equivalent force-couple system at G.arrow_forwardA 20 lb horizontal force P acts on a bell crank as shown below. (a) Replace P with an equivalent force-couple system at B. (b) Find the two vertical forces at C and D which are equivalent to the couple found in part aarrow_forwardNot use ai pleasearrow_forward
- Sociobiology and crime.arrow_forwardConsider the linear system whose augmented matrix is [1 1 1 1 1] 1 2 3 4 5 Identify which variables are basic and which variables are free, and then give the general solution of this system by expressing the basic variables in terms of the free variables. Show all of your work.arrow_forward-0-3-0-8 10 = = 5 Determine if w can be written as a linear combination of V1, V2, and v3. Show all of your work and justify your conclusions.arrow_forward
- Q1.2 1 Point Which of the following best describes Span O a point two points a line O a plane O all of R³ Save Answer Q1.3 1 Point Which of the following best describes Span O a point two points a line O a plane O all of R³ Save Answer Q1.4 1 Point Which of the following best describes Span O a point O three points a line O a plane O all of R³ Save Answerarrow_forward3/5 as a percentagearrow_forward1 2/3 divided by 3/4arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
![Text book image](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781938168383/9781938168383_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781337282291/9781337282291_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)