Does the matrix 1 −1 -2 1 0 -1 have an LU factorization? If so, compute it. If not, explain why not. B = 2 -4 3
Does the matrix 1 −1 -2 1 0 -1 have an LU factorization? If so, compute it. If not, explain why not. B = 2 -4 3
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
![**Matrix LU Factorization Problem**
**Problem Statement:**
b) Does the matrix
\[
B =
\begin{bmatrix}
2 & 1 & -1 \\
-4 & -2 & 1 \\
3 & 0 & -1
\end{bmatrix}
\]
have an LU factorization? If so, compute it. If not, explain why not.
**Explanation:**
To determine if the matrix \( B \) has an LU factorization, we need to check if it can be decomposed into a lower triangular matrix \( L \) and an upper triangular matrix \( U \) such that \( B = LU \).
- **LU Factorization:** This is possible when the matrix can be transformed by operations that do not require row exchanges (i.e., without pivoting).
- **Steps:**
1. Attempt to transform the matrix into the required triangular forms.
2. Check if any pivot (leading diagonal element) is zero at any stage, as this would require pivoting, making simple LU decomposition impossible without row exchanges.
After verifying these steps, proceed to compute matrices \( L \) and \( U \), if possible, or explain why it is not feasible.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed3eef2a-70e8-485e-b91d-563de65921b3%2F6a372463-f9fe-4a82-aba1-a221ab56eb60%2Fcxi91j7_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix LU Factorization Problem**
**Problem Statement:**
b) Does the matrix
\[
B =
\begin{bmatrix}
2 & 1 & -1 \\
-4 & -2 & 1 \\
3 & 0 & -1
\end{bmatrix}
\]
have an LU factorization? If so, compute it. If not, explain why not.
**Explanation:**
To determine if the matrix \( B \) has an LU factorization, we need to check if it can be decomposed into a lower triangular matrix \( L \) and an upper triangular matrix \( U \) such that \( B = LU \).
- **LU Factorization:** This is possible when the matrix can be transformed by operations that do not require row exchanges (i.e., without pivoting).
- **Steps:**
1. Attempt to transform the matrix into the required triangular forms.
2. Check if any pivot (leading diagonal element) is zero at any stage, as this would require pivoting, making simple LU decomposition impossible without row exchanges.
After verifying these steps, proceed to compute matrices \( L \) and \( U \), if possible, or explain why it is not feasible.
Expert Solution
Step 1
We have given a matrix,
A square matrix is said to have an factorization if it can be written as the product of lower triangular and upper triangular matrix.
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