### LU Decomposition (Factorization) - Educational Content#### LU DecompositionLU decomposition involves factoring a matrix $ A $ into the product of a lower triangular matrix $ L $ and an upper triangular matrix $ U $.#### Problem StatementFind an LU decomposition (factorization) of:\[A = \begin{bmatrix} 8 & 9 \\ -5 & 3 \end{bmatrix}\]That is, write $ A = LU $ where $ L $ is a lower triangular matrix with ones in the diagonal, and $ U $ is an upper triangular matrix.#### Application of LU DecompositionUse the LU factorization to solve the system:\[\begin{bmatrix} 8 & 9 \\ -5 & 3 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 10 \\ 11 \end{bmatrix}\]#### Step-by-Step Solution Process1. **LU Factorization:** \[ A = LU = \begin{bmatrix} 1 & 0 \\ l_{21} & 1 \end{bmatrix} \begin{bmatrix} u_{11} & u_{12} \\ 0 & u_{22} \end{bmatrix} \]2. **Construct matrices $ L $ and $ U $ such that:** \[ \begin{bmatrix} 8 & 9 \\ -5 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ l_{21} & 1 \end{bmatrix} \begin{bmatrix} u_{11} & u_{12} \\ 0 & u_{22} \end{bmatrix} \] - From the first row: $u_{11} = 8$ and $ u_{12} = 9 $. - From the element $ -5 = l_{21} \times 8 $ we get \( l_{21} = -\frac{5}{8}

Answered: Image /qna-images/answer/015674c8-0a31-4bdc-851f-7f489ae8f156.jpg

Find an LU decomposition (factorization) of 9. 8 A = 5 3 That is, write A = LU where L is a lower triangular matrix with ones in the diagonal, and U is an uppe triangular matrix. Use the LU factorization to solve the system 8. 91 10 -5 3 x2 11 A = LU = 18 云=

Find an LU decomposition (factorization) of 9. 8 A = 5 3 That is, write A = LU where L is a lower triangular matrix with ones in the diagonal, and U is an uppe triangular matrix. Use the LU factorization to solve the system 8. 91 10 -5 3 x2 11 A = LU = 18 云=

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### LU Decomposition (Factorization) - Educational Content

#### LU Decomposition

LU decomposition involves factoring a matrix $ A $ into the product of a lower triangular matrix $ L $ and an upper triangular matrix $ U $.

#### Problem Statement

Find an LU decomposition (factorization) of:

\[
A = \begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix}
\]

That is, write $ A = LU $ where $ L $ is a lower triangular matrix with ones in the diagonal, and $ U $ is an upper triangular matrix.

#### Application of LU Decomposition

Use the LU factorization to solve the system:

\[
\begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
10 \\
11
\end{bmatrix}
\]

#### Step-by-Step Solution Process

1. **LU Factorization:**

\[
A = LU = \begin{bmatrix}
1 & 0 \\
l_{21} & 1
\end{bmatrix}
\begin{bmatrix}
u_{11} & u_{12} \\
0 & u_{22}
\end{bmatrix}
\]

2. **Construct matrices $ L $ and $ U $ such that:**

\[
\begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix} =
\begin{bmatrix}
1 & 0 \\
l_{21} & 1
\end{bmatrix}
\begin{bmatrix}
u_{11} & u_{12} \\
0 & u_{22}
\end{bmatrix}
\]

- From the first row: $u_{11} = 8$ and $ u_{12} = 9 $.
- From the element $ -5 = l_{21} \times 8 $ we get \( l_{21} = -\frac{5}{8}

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