## Finding LU Factorization of a Matrix (with L unit lower triangular)### Problem StatementFind an LU factorization of the matrix $ A $ where $ L $ is a unit lower triangular matrix.The matrix $ A $ is given by:\[ A = \begin{pmatrix} 6 & 4 \\12 & 5 \end{pmatrix} \]### ExplanationLU factorization of a matrix involves decomposing the matrix $ A $ into two matrices: $ L $ and $ U $.- $ L $ is a lower triangular matrix with ones on the diagonal.- $ U $ is an upper triangular matrix.In this case, since $ L $ is a unit lower triangular matrix, it will have ones on the diagonal. The goal is to express $ A $ in the form $ A = LU $, where:\[ L = \begin{pmatrix} 1 & 0 \\l_{21} & 1 \end{pmatrix} \]\[ U = \begin{pmatrix} u_{11} & u_{12} \\0 & u_{22} \end{pmatrix} \]Let's find the values of $ l_{21} $, $ u_{11} $, $ u_{12} $, and $ u_{22} $.1. **Determine $ u_{11} $:** From the first row of $ A $, we have: \[ u_{11} = 6 \] 2. **Determine $ u_{12} $:** Continuing from the first row of $ A $, we get: \[ u_{12} = 4 \]3. **Determine $ l_{21} $:** From the second row of $ A $, since $ l_{21} \times u_{11} = 12 $, we have: \[ l_{21} = \frac{12}{6} = 2 \]4. **Determine $ u_{22} $:** From the second row of $ A $, substituting known values: \[ 2 \times 4 + u_{22} = 5 \Rightarrow u_{22} = 5 -

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Math

Advanced Math

Find an LU factorization of the matrix A (with L unit lower triangular). 6 4 A= 12 5

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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Question

## Finding LU Factorization of a Matrix (with L unit lower triangular)

### Problem Statement
Find an LU factorization of the matrix $ A $ where $ L $ is a unit lower triangular matrix.

The matrix $ A $ is given by:
\[
A = \begin{pmatrix}
6 & 4 \\
12 & 5
\end{pmatrix}
\]

### Explanation
LU factorization of a matrix involves decomposing the matrix $ A $ into two matrices: $ L $ and $ U $.

- $ L $ is a lower triangular matrix with ones on the diagonal.
- $ U $ is an upper triangular matrix.

In this case, since $ L $ is a unit lower triangular matrix, it will have ones on the diagonal. The goal is to express $ A $ in the form $ A = LU $, where:
\[
L = \begin{pmatrix}
1 & 0 \\
l_{21} & 1
\end{pmatrix}
\]
\[
U = \begin{pmatrix}
u_{11} & u_{12} \\
0 & u_{22}
\end{pmatrix}
\]

Let's find the values of $ l_{21} $, $ u_{11} $, $ u_{12} $, and $ u_{22} $.

1. **Determine $ u_{11} $:**
From the first row of $ A $, we have:
\[
u_{11} = 6
\]

2. **Determine $ u_{12} $:**
Continuing from the first row of $ A $, we get:
\[
u_{12} = 4
\]

3. **Determine $ l_{21} $:**
From the second row of $ A $, since $ l_{21} \times u_{11} = 12 $, we have:
\[
l_{21} = \frac{12}{6} = 2
\]

4. **Determine $ u_{22} $:**
From the second row of $ A $, substituting known values:
\[
2 \times 4 + u_{22} = 5 \Rightarrow u_{22} = 5 -

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