108 : 0 0 1 0 A B : 0 : 0 x y z : b (a): Explain specific values of A and B such that the corresponding system of equations has one unique solution. If no such values can be found, explain why not. (b): Explain specific values of A and B such that the corresponding system of equations has infinitely many solutions. If no such values can be found, explain why not. (c): Explain specific values of A and B such that the corresponding system of equations has no solution. If no such values can be found, explain why not.
108 : 0 0 1 0 A B : 0 : 0 x y z : b (a): Explain specific values of A and B such that the corresponding system of equations has one unique solution. If no such values can be found, explain why not. (b): Explain specific values of A and B such that the corresponding system of equations has infinitely many solutions. If no such values can be found, explain why not. (c): Explain specific values of A and B such that the corresponding system of equations has no solution. If no such values can be found, explain why not.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Consider the matrix below that represents a homogeneous system, and has been reduced:
![### Matrix System Analysis
Consider the following augmented matrix:
\[
\begin{pmatrix}
1 & 0 & 8 & \vert & 10 \\
0 & 1 & 0 & \vert & 0 \\
0 & A & B & \vert & b \\
\end{pmatrix}
\]
This matrix corresponds to a system of linear equations involving variables \( x \), \( y \), and \( z \).
**Tasks:**
(a) **Explain** specific values of \( A \) and \( B \) such that the corresponding system of equations has one unique solution. If no such values can be found, **explain** why not.
(b) **Explain** specific values of \( A \) and \( B \) such that the corresponding system of equations has infinitely many solutions. If no such values can be found, **explain** why not.
(c) **Explain** specific values of \( A \) and \( B \) such that the corresponding system of equations has no solution. If no such values can be found, **explain** why not.
### Problem Explanation
To determine what specific combinations of \( A \) and \( B \) lead to different types of solutions, we need to consider the conditions that affect the rank of the matrix and the implications for consistency and the number of solutions:
- **Unique Solution:** For a unique solution, the system must be consistent and the rank of the matrix must equal the number of variables.
- **Infinitely Many Solutions:** The system is consistent, but the rank of the matrix is less than the number of variables, or there are free variables.
- **No Solution:** The system is inconsistent, indicated by a contradictory equation such as 0 = b where b is a non-zero constant.
Understanding these principles will guide you in assigning values to \( A \) and \( B \) to obtain the specified outcomes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4eacb38d-0df4-4d82-94b7-88634a2db476%2F4f92d317-a8ae-45aa-81b7-3fcadc33adad%2Ffil89uj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix System Analysis
Consider the following augmented matrix:
\[
\begin{pmatrix}
1 & 0 & 8 & \vert & 10 \\
0 & 1 & 0 & \vert & 0 \\
0 & A & B & \vert & b \\
\end{pmatrix}
\]
This matrix corresponds to a system of linear equations involving variables \( x \), \( y \), and \( z \).
**Tasks:**
(a) **Explain** specific values of \( A \) and \( B \) such that the corresponding system of equations has one unique solution. If no such values can be found, **explain** why not.
(b) **Explain** specific values of \( A \) and \( B \) such that the corresponding system of equations has infinitely many solutions. If no such values can be found, **explain** why not.
(c) **Explain** specific values of \( A \) and \( B \) such that the corresponding system of equations has no solution. If no such values can be found, **explain** why not.
### Problem Explanation
To determine what specific combinations of \( A \) and \( B \) lead to different types of solutions, we need to consider the conditions that affect the rank of the matrix and the implications for consistency and the number of solutions:
- **Unique Solution:** For a unique solution, the system must be consistent and the rank of the matrix must equal the number of variables.
- **Infinitely Many Solutions:** The system is consistent, but the rank of the matrix is less than the number of variables, or there are free variables.
- **No Solution:** The system is inconsistent, indicated by a contradictory equation such as 0 = b where b is a non-zero constant.
Understanding these principles will guide you in assigning values to \( A \) and \( B \) to obtain the specified outcomes.
Expert Solution
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Step 1
Given matrix in reduced form is
for the system
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