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=
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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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}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F088399be-9a43-44b3-a949-bec438b8f98b%2F015674c8-0a31-4bdc-851f-7f489ae8f156%2F0s78v9i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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