Q62E Consider the matrix A=[11-132-52... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q62E (page 234)

Consider the matrix $A = [\begin{matrix} 1 & 1 & - 1 \\ 3 & 2 & - 5 \\ 2 & 2 & 0 \end{matrix}]$ with LDU-factorizationrole="math" localid="1660129574693" $A = [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}]$ . Find the LDU-factorization for $A^{T}$ .

Short Answer

Expert verified

The LDU-factorization of $A^{T}$ is $[\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ - 1 & 2 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$ .

Step by step solution

Determine the LDU-factorization.

Consider the matrix $A = [\begin{matrix} 1 & 1 & - 1 \\ 3 & 2 & - 5 \\ 2 & 2 & 0 \end{matrix}]$ and LDU-factorization.

$A = [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}]$

Take transpose both side in the equation role="math" localid="1660129618004" $A = [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}]$ as follows.

$A = [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}] A^{T} = {([\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}])}^{T} A^{T} = {([[\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}]])}^{T} A^{T} = ({\{[\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}]\}}^{T})$

Further, simplify the equation as follows.

role="math" localid="1660129922977" $A^{T} = ({\{[\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}]\}}^{T}) A^{T} = ([\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}] {[\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 0 & 0 \end{matrix}]}^{T}) A^{T} = [\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ - 1 & 2 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$

Hence, the LDU-factorization of $A^{T}$ is $[\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ - 1 & 2 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 1 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Determine the LDU-factorization.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Applied Mathematics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Consider the matrix A=[11-132-5220] with LDU-factorizationrole="math" localid="1660129574693" A=[100310200][1000-10002][11-1012001]. Find the LDU-factorization forAT.

Short Answer

Step by step solution

Determine the LDU-factorization.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Applied Mathematics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.