A linear system is inconsistent if and only if its augmented matrix has a row of the form [00... 0 h] where h‡ 0. True False
A linear system is inconsistent if and only if its augmented matrix has a row of the form [00... 0 h] where h‡ 0. True False
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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![### Understanding Inconsistent Linear Systems
**Statement**:
"A linear system is inconsistent if and only if its augmented matrix has a row of the form \([0 \; 0 \; \ldots \; 0 \; h]\), where \(h \neq 0\)."
**Explanation**:
- This means that for a linear system of equations, inconsistency occurs when, during row operations, we obtain a row where all the coefficients of the variables are zero, but the constant term (on the right side of the augmented matrix) is non-zero.
- **Matrix Row Representation**:
\[
[0 \; 0 \; \ldots \; 0 \; h]
\]
- This row indicates that \(0x_1 + 0x_2 + \ldots + 0x_n = h\). In simpler terms, it implies \(0 = h\).
- Since \(h \neq 0\), this creates a contradiction, meaning the equations cannot be satisfied simultaneously, leading to an inconsistency in the system.
**Question**:
Is the statement "A linear system is inconsistent if and only if its augmented matrix has a row of the form \([0 \; 0 \; \ldots \; 0 \; h]\), where \(h \neq 0\)" true or false?
- True
- False
### Answer:
For a deeper understanding, reflect on the nature of linear equations and how row reduction can reveal inconsistencies. When all the variables cancel out, and we are left with a non-zero constant, it clearly indicates that no solution exists for the system, verifying the statement above as **True**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc353cc0f-0c4d-4d5b-a3e6-e746dbb90f94%2F42d5a790-5590-4254-8629-70612f650d73%2Fszvn24g_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Inconsistent Linear Systems
**Statement**:
"A linear system is inconsistent if and only if its augmented matrix has a row of the form \([0 \; 0 \; \ldots \; 0 \; h]\), where \(h \neq 0\)."
**Explanation**:
- This means that for a linear system of equations, inconsistency occurs when, during row operations, we obtain a row where all the coefficients of the variables are zero, but the constant term (on the right side of the augmented matrix) is non-zero.
- **Matrix Row Representation**:
\[
[0 \; 0 \; \ldots \; 0 \; h]
\]
- This row indicates that \(0x_1 + 0x_2 + \ldots + 0x_n = h\). In simpler terms, it implies \(0 = h\).
- Since \(h \neq 0\), this creates a contradiction, meaning the equations cannot be satisfied simultaneously, leading to an inconsistency in the system.
**Question**:
Is the statement "A linear system is inconsistent if and only if its augmented matrix has a row of the form \([0 \; 0 \; \ldots \; 0 \; h]\), where \(h \neq 0\)" true or false?
- True
- False
### Answer:
For a deeper understanding, reflect on the nature of linear equations and how row reduction can reveal inconsistencies. When all the variables cancel out, and we are left with a non-zero constant, it clearly indicates that no solution exists for the system, verifying the statement above as **True**.
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