### Solving Linear Systems Using LU Factorization#### Problem StatementSolve the linear system $ \mathbf{A}\mathbf{x} = \mathbf{b} $ by finding an $ LU $-factorization of the coefficient matrix $\mathbf{A}$, solving the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $, and solving the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.Given system:\[ \begin{aligned}2x + y &= 1 \\y - z &= 2 \\-2x + y + z &= -2 \\\end{aligned}\]#### Steps to Solve**(a) Find an $ LU $-factorization of the coefficient matrix $\mathbf{A}$.** Your $ \mathbf{L} $ matrix must be unit diagonal.The process involves decomposing matrix $\mathbf{A}$ into a lower triangular matrix $\mathbf{L}$ and an upper triangular matrix $\mathbf{U}$.\[LU = \begin{bmatrix} & & \\ & & \\ & & \end{bmatrix}\rightarrow\begin{bmatrix} & & \\ & & \\ & & \end{bmatrix}\rightarrow\begin{bmatrix} & & \\ & & \\ & & \end{bmatrix}\]**(b) Solve the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $.** \[y = \begin{bmatrix} \\ \\ \end{bmatrix}\rightarrow\]**(c) Solve the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.** \[x = \begin{bmatrix} \\ \\ \end{bmatrix}\rightarrow\]### Diagram Explanation1. **LU Decomposition:** - **Matrix $\mathbf{L}$**: A 3x3 lower triangular matrix with a unit diagonal (diagonal elements are 1). - **Matrix $\mathbf{U}$**: A 3x3 upper

Answered: Image /qna-images/answer/021e22c4-0c6d-42f4-a55a-e468d5371445.jpg

Solve the linear system Ax = b by finding an LU-factorization of the coefficient matrix A, solving the lower triangular system Ly = b, and solving the upper triangular system Ux = y. 2x + y 1 %3! y - z = -2x + y + z = -2 (a) Find an LU-factorization of the coefficient matrix A. (Your L matrix must be unit diagonal.) LU = (b) Solve the lower triangular system Ly = b. y = (c) Solve the upper triangular system Ux = y. X = D00

Solve the linear system Ax = b by finding an LU-factorization of the coefficient matrix A, solving the lower triangular system Ly = b, and solving the upper triangular system Ux = y. 2x + y 1 %3! y - z = -2x + y + z = -2 (a) Find an LU-factorization of the coefficient matrix A. (Your L matrix must be unit diagonal.) LU = (b) Solve the lower triangular system Ly = b. y = (c) Solve the upper triangular system Ux = y. X = D00

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Solve the linear system by its augmented matrix. X1 + 2x2-x3-2x4 = 4 x1 + 3x2+2x3-4x4 = 6…

Q: Which matrix represents the system of equations below? [-2x-7y=3 7x-6y=12

A: System of equation

Q: Fuse an inverse matrix to solve the given system of equations. X-X, +x, = 1 2x, +x, +2x, =2 3x, +2x2…

Q: 2х + Зу + 6z 3D 3 B. Solve the linear system-x + 2y + 2z = -1 Зх — 4у - 5z %3 -6

Q: х, — 2х, + 3х, 3D1 2х, — 3х, — 2х, %3D3 2x, х, — х, — 5х, %3D 2

Q: a) Use the inverse of a matrix to solve the linear system: X1 - X2 - X3 = 5 X2 + x3 = -1 . (x1-2x2 -…

A: Dear student you have asked multiple questions. As per our policy your first question will be…

Q: X1 — 5х, + 2х; - 20 - —Зx1 + х2 — Хз %3D 8 — 2х2 + 5хз -16.

A: Given system of linear equation is x1−5x2+2x3=−20−3x1+x2−x3=8−2x2+5x3=−16..........1 Write the given…

Q: Consider the following system of linear equations for four real variables x1, x2, X3, X4: x2 + x3 –…

A: x2+x3-2x4=-3x1+2x2-x3=22x1+4x2+x3-3x4=-2x1-4x2-7x3-x4=-1011-212-10241-31-4-7-1x1x2x3x4=-32-2-19Let…

Q: Write the system of linear equations in the form Ax = b and solve this matrix e 5x2 + 2x3 = -13 x2-…

A: Given

Q: 1) Solve the system by using an LU-factorization of the coefficient matrix. -4 x, + 6x, = –10 x, -…

A: Given system is -4x1+6x2=-10 x1-3x2=1

Q: Use the inverse of the coefficient matrix to solve the system of equations. x+2y+3z = 16 2x+3y + 2z…

Q: Solve the system of equations {−9x+9y=360 {-5x-4y=-529 by converting to a matrix equation and using…

Q: 2x+y=z=7 -3x + 2y + z = -1 x + 3z = 0 and remembering our equation AX = B, answer the following…

A: Given: To Find: (a) To write down (b) To determine whether exists and to find it.(c) To solve the…

Q: m of linear equations x +y = 3z = 2 2x + 3y + 2 = 0 3x + 4y = 2 =1 using 23 the

Question

### Solving Linear Systems Using LU Factorization

#### Problem Statement
Solve the linear system $ \mathbf{A}\mathbf{x} = \mathbf{b} $ by finding an $ LU $-factorization of the coefficient matrix $\mathbf{A}$, solving the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $, and solving the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.

Given system:
\[
\begin{aligned}
2x + y &= 1 \\
y - z &= 2 \\
-2x + y + z &= -2 \\
\end{aligned}
\]

#### Steps to Solve

**(a) Find an $ LU $-factorization of the coefficient matrix $\mathbf{A}$.**

Your $ \mathbf{L} $ matrix must be unit diagonal.

The process involves decomposing matrix $\mathbf{A}$ into a lower triangular matrix $\mathbf{L}$ and an upper triangular matrix $\mathbf{U}$.

\[
LU =
\begin{bmatrix}
& & \\
& & \\
& &
\end{bmatrix}
\rightarrow
\begin{bmatrix}
& & \\
& & \\
& &
\end{bmatrix}
\rightarrow
\begin{bmatrix}
& & \\
& & \\
& &
\end{bmatrix}
\]

**(b) Solve the lower triangular system $ \mathbf{L}\mathbf{y} = \mathbf{b} $.**

\[
y =
\begin{bmatrix}
\\
\\

\end{bmatrix}
\rightarrow
\]

**(c) Solve the upper triangular system $ \mathbf{U}\mathbf{x} = \mathbf{y} $.**

\[
x =
\begin{bmatrix}
\\
\\

\end{bmatrix}
\rightarrow
\]

### Diagram Explanation

1. **LU Decomposition:**
- **Matrix $\mathbf{L}$**: A 3x3 lower triangular matrix with a unit diagonal (diagonal elements are 1).
- **Matrix $\mathbf{U}$**: A 3x3 upper

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Trending now

This is a popular solution!

Step by step

Solved in 5 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Matrix Factorization$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Use a software program or a graphing utility to solve the system of linear equations using an inverse matrix.
Write the system of linear equations in the form Ax = b and solve this matrix equation for x. -2x1 3x₂ = -21 6x1 + x₂ = -9 Q-E Need Help? X1 Read It -21 RHE -9
I need help with part a and part c. Thank you!
Use an augmented matrix to find the solution of the system of equations. + 7y + 62 = -7 2y 2 = 19 2y + 22 = 1 X -5x 5x x = y = 2 =
Solve the system of equations using augmented matrix methods x¹-2x²=-3 2x¹-x²=6